Given the mean and standard deviation of the normally distributed changes in two measurements from before and after treatment from the same population, is it possible to estimate the covariance of the two samples?
Assume there two things that are measured from a given population before and after some treatment.
However, instead of having the measurements of each individual in the population, we have the distributions of the changes from before and after treatment.
So the differences before and after would be:
$$\delta X = X_{end} - X_{beg} \\$$ $$\delta Y = Y_{end} - Y_{beg} \\$$
The data available, however, are the parameters of the distributions of the changes, $X_{\delta}$ & $Y_{\delta}$, i.e. $mean(\delta X), std(\delta X)$ & $mean(\delta Y), std(\delta Y)$
Where:
$$X_{\delta} \sim \mathcal{N}(mean(\delta X), std(\delta X)) \\$$ $$Y_{\delta} \sim \mathcal{N}(mean(\delta Y), std(\delta Y)) \\$$
In other words, given only the parameters of the distributions of the changes in the the samples, and not the actual individual samples, is there some way to estimate the covariance?
No. Estimation of covariance requires some information about how the two groups vary together. In your case, the "before" and "after" measurements constitute the two groups, but without knowing how the "before" and "after" measurements are paired up for each individual, it is not possible to determine covariance.