You should mostly just ensure that the requests your software makes are correct, and that you're testing response handling with correct responses; what actually happens at the other end is of no concern to you.

At some point, of course, you may want to run actual integration (preferably against test users/whatever) but that needn't be a part of your regular test suite, so you can for example add an env toggle to actually perform remote connections, or have a separate normally-excluded set of tests for that.