Well, we use the Present Continuous to talk about something in the future that we have agreed to do:
Tim and I are going to the theatre tonight, then we're having dinner at White's.
Are you doing anything at the weekend? - I'm playing tennis with Mary on Saturday.
So, I'd say it's OK to use the Present Continuous in your sentence about watching TV at home. Using the Present Continuous, you show your agreement to do it together.
But your teacher is right too. Of course, we can say "Mike and I are going to watch a film." But using to be going to, we talk about our plans, rather than arrangements.
There are many grammar books that deal with the issue (Present Continuous vs. to be going to for future events). But the explanation I have given is inspired by Oxford Living Grammar intermediate.