It depends on the circumstances - what is being sampled how (i.e. what kind of relationships should exist between observations in the two regressions)?
If you're prepared to assume the observations in the two regressions are independent of each other it's quite straightforward -- the covariance is zero. However the formula in your question is wrong; you need to square the standard errors under the square root (and it should be made clear that the $b$'s are estimates, not population coefficients).
If at least some observations are pairwise dependent across the regressions, you're essentially in a version of a seemingly unrelated regressions style of problem (as you already anticipate).