"At my training" is fine.
Training is a noun, and like many nouns for activities (such as the names of sports) it is very flexible and can refer to a single session, a place you go to for the activity, or an activity you attend regularly. For example, "I'm going to football" could mean you are going to the place you regularly go to play football; likewise, "I'm at football" could mean you are at the place you go to play football.
The use of "my" suggests that the person you are speaking to is already aware that you are taking part in some training. If you add "session" to the end, it sounds like you have previously made them aware of a particular session, or perhaps that the training only consists of one session.
Your alternative "in my training" is also fine, but would mean you are actually engaged in the session at that moment, rather than just "at" the place it is taking place. "In training" does have another use, which can refer to being engaged in ongoing training (not necessarily at that moment), for example "I'm in training to be a teacher".