Generally speaking, these sorts of relationships are encouraged. People like to see collaboration, not only from a scientific perspective, but from a networking and professional development standpoint.
Usually, nothing is needed to join a lab, other than a willingness on your part to participate. If you want a mentor-mentee relationship with an advisor who is not in your department (i.e. the department you were admitted to), that is something you will have to negotiate with them. As I mentioned in the comments, students in the advisor's own department are their first priority for mentorship and time.
You say in the comments:
I would have to verify this, but my primary advisor may officially be in my dept, but I would be working more closely with prof. outside my dept.
While possible, I just want to warn you that this is probably not the most likely outcome, although it's possible standards in your fields may be more permissible.
You should also keep in mind obligations to your home department. If you are funded on an RAship (or an internal fellowship, likely), then you are expected to devote 20 hours/week to your funder's lab, not leaving you much time to work for the other advisor. Of course, the other advisor may fund you, but again, it is probably hard to devote a line of funding to a student outside of the department. Your primary advisor may make your duties including working for the other professor, but I can't tell you how likely that is.
If you are on an external fellowship, your time is much more flexible.
Tl;dr: Having a primary mentor outside of your home department is probably not likely, but a good mentor would help you arrange something to the best of their ability, as long as you are a good communicator with all three parties.