I've tackled carpenter ants a few times, and have now settled on my favorite Boric Acid recipe, but my understanding is that termites are a completely different ball game requiring a much more noxious attack.
Your professional exterminator SHOULD be the guy to answer the questions concerning dangers and a game plan, but I wouldn't have much confidence in him at this point.
Until you find out exactly what pesticides were used and if they are meant to be used for both indoor/outdoor applications, and a safe time line for the fumes to become inert or cleaned up, I'd be leary about breathing in the fumes without some type of protection and letting others breath it in as well.
You'd hate to have a potential buyer look at the property, then dream up some lawsuit just to put the icing on the cake.
If termites are a well known issue in your area and the smell of the pesticide is noticeable, you could always get the agent to put a positive spin on it, that you had extra applications applied to prevent problems for any future owners
I hope it resolves well and easily for you.