I teach English at an elementary school in Japan. While working I normally call the teachers by their last name plus 先生. Like 田中先生 for example.
However, after work, let's say 田中先生 invites me to have a drink at an 居酒屋。Am I still expected to call him (or her)「先生」 even if it's just the two of us and we're outside of school? Would it be rude or inappropriate to call him 田中さん? Does the teacher's age or gender make a big difference?
Edit: Thanks for the replies! If I was Professor Moriarty's student, I would call him Professor. If he was a co-worker that I've known for years and consider a friend, I imagine at some point I would start calling him by his first name. In America, I think most people let a person know when they can use their first name or nickname by directly saying something like "Call me Bob!".
How about Japan? When a Japanese person becomes friends with someone, do they usually say, "You can call me ...!"? Switching from first names to last names or knowing when I can drop さん or 先生 has always been confusing for me.
Do you think it would be OK if I just asked directly? Like, "Can I call you ...?"
To answer some of the questions asked,
I am an ALT. I've been teaching at this school for 2 years. I normally try to use 敬語 and always use です and ます。 Most of the teachers speak informally to me and don't use much 敬語。But most of the teachers are a lot older than me, so I'm not sure if it's because I'm younger or if it's an invitation for me to speak to them casually.