Using incomplete regression output to determine the value of the test statistic I have incomplete regression output subsisting of the sample size, the standard error of the regression coefficient, and the p-value for two variables in a regression model. Can I determine what the regression coefficients are? Can I infer which coefficient indicates a larger value for the regression coefficient?
 A: I'm going to cover how the p-value is calculated first, then how to work backwards from the p-value. 
The t-statistic used to compute a p-value for a regression coefficient is $t= {\beta \over SE}$
Then the usually reported p-value is calculated from $|t|$ and a t-distribution with $n-k$ degrees of freedom, $n$ being sample size and $k$ the number of coefficients including intercept. Given $|t|$, as AdamO states, you won't be able to recover the sign of the coefficient, just its magnitude.
You could do this calculation in R with the pt() function to calculate the probability (multiplying by 2 since it is a two sided test):
2 * (1 - pt(beta/se, n-k)) 
But here you already have the p-value, and need to work backwards. You can go from a probability to the t-statistic $|t|$ using the qt() function (dividing p-value by 2 due to two sided):
abs(qt(p / 2, n-k))
From there, calculating $|\beta|$ is straightforward.
Example with $\beta=0.72727$, $SE = 0.44998$, $n=6$, $k=3$
> 2*(1-pt(0.72727/0.44998,df=3))
[1] 0.2044641 
> qt(0.2044641/2,3)
[1] -1.616227
> abs(qt(0.2044641/2,3))*.44998
[1] 0.7272698
A: If the p-values come from a two-tailed test, then you're out of luck. There's no way to determine whether a certain $\hat{\beta}_2$ having a very small p-value relative to $\hat{\beta}_1$ having the same standard errors indicates that it's a very large positive slope or a very large negative slope. However, since the inference from the p-value indicates whether or not the $\beta$s are equal to zero, you can reverse transform the p-value using the cumulative normal distribution to find its z-score (Wald test statistic), and multiply the z-score by the standard error to obtain the absolute value of the regression coefficient. With assumptions on the ordering (or perhaps you were interested in the regression coefficient with the largest absolute value), you can answer your questions with these reverse transformed regression output values.
