I've found out that for my original answer below I misunderstood the setting. Here is something regarding testing the equality of two coefficients $\beta_1$ and $\beta_2$ in a regression. You will need to use something that is called "general linear hypothesis" in linear regression for testing $\beta_1-\beta_2=0$. I will not look this up for you because my time is limited and I had answered originally thinking that I could say something without doing some reading to remind myself.
Regarding $\beta_1$ and $\beta_3$ from two different regressions, it is important to model the dependence between the regressions, which is probably best done running a multivariate regression with outputs $Y_1$ and $Y_2$ and all four variables $V_1$, $V_2$, $V_3$, $V_4$. The again a linear hypothesis regarding $\beta_1-\beta_3$ can be tested.
The paired t-test doesn't address this problem because there is no sample of independent pairs here.
The following answer was written based on wrong understanding of the problem. I leave it here in case anybody has the problem addressed here:
Response in comment says: "I would have one pair (e.g., $\beta_1$ and $\beta_2$) for each person". Assuming that the persons don't influence each other and the $\beta$ are computed separately for each person not involving other persons' data, yes, you could take the $\beta_1,\beta_2$ for each person as a pair to be analysed by a paired t-test (have a look at your data whether there are outliers or strong skewness in the differences between the $\beta$ though). I'm also here assuming that the null hypothesis that you want to test is $\beta_1=\beta_2$. The model then implies that there are random coefficients $\beta_{1i}, \beta_{2i}$ for each person $i$ with a fixed expected value of $\beta_{1i}-\beta_{2i}$, zero under the null hypothesis. This may make sense, although I obviously don't know the details of what you're doing.
Note that this involves a dependence (namely between $\beta_1$ and $\beta_2$ of the same person) and also an independence assumption (between different persons).
I'm not sure what exactly you mean by "looking at confidence intervals of coefficients" - you mean for every single person? This will give you twice as many confidence intervals as you have persons (as long as you're only looking at $\beta_1$ and $\beta_2$) - I'm not quite sure how you'd interpret that. Also these confidence intervals will not take dependence between the different $\beta$-coefficients into account.
"How would I conduct the respective statistic?" You'd just run it, with the $\beta_{1i}, \beta_{2i}$ for each person $i$ as paired values (which actually amounts to running a one-sample t-test on $\beta_{1i}-\beta_{2i}$).
As long as there is no specific knowledge on how your two regressions within a person depend on each other (or not), the same holds for comparing $\beta_1$ and $\beta_3$ etc.
