If I get a pet door installed for my dog, does it go both ways?

Yes! They are meant for letting pets in and out. There are some cat doors such as the Ani Mate 234 and 235 that have a 4-way locking feature allowing you to manually control access. This lock will allow you to leave the door open so they can enter and exit, lock it totally so they can’t go in or out, make it in only so once they come in they can go back outside, and the opposite of that out only so once they go out they can’t come back in. Source:

Yes the pet door goes both ways. I have one for my dog she loves it.

Yes otherwise it would be like a trap door if it's one way

I think it does you can check out before installation though

Ours goes in and out. Our cat can go in and out of the garage as much as he likes. Unfortunately so can a few other cats around here!

Well, since the whole point of a doggy door is to let the dog in and out of the house use the bathroom or to just roam or lay in the sun I would say yes all pet doors open inward and outward. It would kind of be pointless to have one that won't let the dog back in the house after it is done outside or won't let it outside at all. The only thing with these things is that if it can let the dog/cat in and out then it can let other animals also in and out of your house.