Researcher 1 runs 1000 regressions, researcher 2 runs only 1, both get same results -- should they make different inferences? Imagine a researcher is exploring a dataset and runs 1000 different regressions and he finds one interesting relationship among them.
Now imagine another researcher with the same data  runs just 1 regression, and it turns out it's the same one that the other researcher took 1000 regressions to find. Researcher 2 does not know researcher 1.
Should researcher 1 make different inferences than researcher 2? Why? For example, should researcher 1 perform multiple comparisons correction, but researcher 2 should not?
If researcher 2 showed you his single regression first, what inferences would you make? If after that researcher 1 showed you his results, should you change your inference? If so, why should it matter?
PS 1: If talking about hypothetical researchers makes the problem abstract, think about this: imagine you ran just one regression for your paper, using the best method available. Then another researcher explored 1000 different regressions with the same data, until he found the exact same regression you ran. Should you two make different inferences? Is the evidence the same for both cases or not?  Should you change your inference if you knew the other researcher results? How should the public assess the evidence of the two studies? 
PS 2: please try to be specific and to provide a mathematical/theoretical justification, if possible! 
 A: The statistical interpretation is much less clear than, what you are asking for, the mathematical treatment.
Mathematics is about clearly defined problems. E.g. rolling a perfect dice, or drawing balls from an urn. 
Statistics is applied mathematics where the mathematics provides a guideline but is not the (exact) solution.
In this case it is obvious that circumstances play an important role. If we perform a regression and then calculate (mathematics) some p value to express the strength then what is the interpretation (statistics) and value of the p value? 


*

*In the case of the 1000 regressions performed by researcher 1 the result is much more weak since this type of situation occurs when we do not really have a clue and are just exploring the data. The p value is just an indication that there may be something. 
So the p value is obviously less worth in the regression performed by researcher 1. And if researcher 1 or somebody using the results of researcher 1 would like to do something with the regression then the p value needs to be corrected. (and if you thought the difference between researcher 1 and researcher 2 was not enough, just think about the multitude of ways
that researcher 1 can to correct the p value for multiple
comparisons)

*In the case of the single regression performed by researcher 2 the result is much stronger evidence. But that is because the regression does not stand on it's own. We have to include the reasons why researcher 2 did only one single regression. This could be because he had good (additional) reasons to already believe that the single regression is a good model for the data.

*The setting of the regressions performed by researcher 1 and 2 is much different, and it is not often that you encounter both at the same time for the same problem. If this is the case then either 


*

*researcher 2 was very lucky
This is not so uncommon, and we should better correct for this when
interpreting literature, as well we should improve the publishing of
the total picture of research. If there are a thousand researchers
like researcher 2, and we will only see one of them publish a
success, then because we did not see the failures of the other 999
researchers we might mistakingly believe we did not have a case like
researcher 1

*researcher 1 was not so smart and did an incredibly superfluous search for some regression while he might have possibly known from the start that it should have been that single one, and he could have performed a stronger test.
For outsiders who are smarter than researcher 1 (do not care about the additional 999 regressions from the start) and read about the work, they might give more strength to the significance of the results, however still not as strong as he would do for the outcome of researcher 2. 
While researcher 1 may have been too conservative when correcting for 999 superfluous additional regressions, we can not ignore the fact that the research was done in a vacuum of knowledge and it is much more likely to find a lucky researcher of the type 1 than the type 2.
An interesting related story: In astronomy, when  they were planning a better instrument to measure the cosmic background with higher precision, there were researchers that argued to only release half the data. This because there is only one shot to gather data. Once all the regressions have been performed by the dozens of different researchers (and because of the incredible variation and creativity of the theorist, there is certainly some fit to every possible, random, bump in the data), there is no possibility to perform a new experiment to verify (that is, unless you are able to generate a whole new universe).
A: Here's my "Bayesian" slant on your question. I think you have described a situation where two people with different prior information should get a different answer/conclusion when given the same dataset. A more blunt/extreme example is suppose that we have a "researcher 1b" who just happens to guess the regression model parameters and conclusions from whatever hypothesis. Running $1000$ regressions is not conceptually too far away from guessing.
What I think is happening...what do we learn about the researchers prior information from the above question?
   - researcher 1 probably has a flat prior for the models $P (M_k|I_1)=\frac {1}{1000} $
   - researcher 2 has a sharp prior for the model of interest $P (M_1|I_2) =1$ (assume $M_1$ is the model they both fit)
This is obviously a simplification, but you can see here, we already place a lot more weight on researcher 2's inferences without any data.
But you see, once they both take account of the data, researcher 1's posterior probability for $M_1$ will increase... $P (M_1|DI)>>P (M_1|I) $ (...we know this because it was "better" than $999$ other models...). Researcher 2's posterior can't concentrate anymore, it is already equal to $1$.
What we don't know is how much the data supported $M_1$ over the alternatives. 
What we also don't know is how the different models alter the substantive conclusions of researcher 1. For example, suppose all $1000$ models contain a common term, and all $1000$ regression parameters for that variable are significantly greater than $0$ (eg $p-value <10^{-8}$ for all models). Then there is no problem with concluding a significantly positive effect, even though many models were fit.
You also don't say how big the dataset is, and this matters! If you're talking about a dataset with $100$ observations and $10$ covariates/predictors/independent variables, then researcher 1 will probably still be quite uncertain about the model. However, if researcher 1 is using $2,000,000$ observations, this may conclusively determine the model.
There is nothing fundamentally wrong with two people that start with different information, and continue to have different conclusions after seeing the same data. However...seeing the same data will bring them closer together, provided their "model space" overlaps and the data supports this "overlapping region".
A: Short story: we don’t have enough information to answer your question because we don’t know anything about the methods used or the data collected. 
Long answer...The real question here is whether each researcher is doing:


*

*rigorous science

*rigorous pseudoscience

*exploration of data

*data dredging or p-hacking


Their methods will determine the strength of the interpretation of their results. This is because some methods are less sound than others.
In rigorous science we develop a hypothesis, identify confounding variables, develop controls for variables outside our hypothesis, plan test methods, plan our analytical methodology, perform tests / collect data, and then analyze data. (Note that the analytical methods are planned before the test occurs). This is the most rigorous because we must accept data and analysis that does not agree with the hypothesis. It isn’t acceptable to change methods after the fact to get something interesting. Any new hypothesis from the findings have to go through the same process again. 
In pseudoscience we often take data that is already collected. This is more difficult to use ethically because it is easier to add biases to the results. However, it is still possible to follow the scientific method for ethical analysts. It may be difficult to set up proper controls though and that needs to be researched and noted. 
Exploration of data is not based on science. There is no specific hypothesis. There is not a priori evaluation of confounding factors. Also, it is difficult to go back and re-do the analysis using the same data, because the results may be tainted by prior knowledge or modeling and there is no new data to use for validation. A rigorous scientific experiment is recommended to clarify possible relationships found from exploratory analysis. 
Data dredging or P-hacking is where an “analyst” performs multiple tests hoping for an unexpected or unknown answer or manipulates the data to get a result. The results may be simple coincidence, may be the result of confounding variable(s), or may not have have meaningful effect size or power. 
There are some remedies for each problem, but those remedies have to be carefully evaluated. 
