As Nate Eldredge points out in the comments, there are two types of joint appointments:
With a courtesy appointment, you get the public recognition of having both titles (Professor of X and Professor of Y) and you can advise and teach in both departments. However, administratively you have a primary department that pays your salary and sets your teaching/service duties. The main drawback is that if you want to split your teaching between the departments, it may take some arranging. [In my experience this can often be done, but it's not guaranteed by a courtesy appointment.]
With a fully joint appointment, the two departments split your salary and duties in some set fraction (usually, but not necessarily, 50-50). This makes it easy to do substantial amounts of teaching in both departments, but it can mean a slightly higher service load, since a 50% appointment often doesn't really cut the number of meetings in half. The biggest drawback of a joint appointment is having to keep both departments happy, especially if they disagree on how you should be spending your time. [This is a real danger for tenure cases.]
If you'd like a courtesy appointment in another field, you need someone in that department to serve as a champion for your case. You can sound out friends or mentors privately to try to convince someone to advocate for offering you such a position. Once someone brings it up at a department meeting, you then need a good enough reputation that the department is happy to have you associated with it. Ideally, you should be someone they would have loved to hire if you hadn't already joined another department; in practice, you can sometimes get away with a slightly weaker case than that, but you still need to look strong enough that there's no fear you'll hurt the department's reputation. Once the department is behind you, there may also be some administrative hurdles (perhaps including letters of recommendation).
Switching to a genuine joint appointment is much trickier, since you need to convince the other department to start paying for you. Why should they do that when they could offer you a courtesy appointment for free? There are various possible answers, such as:
You're so famous that they'll cheerfully do whatever you want.
They really value you, and offering you this appointment will keep you from leaving.
You'll fill a role in the new department that they would have had to hire someone else for anyway (e.g., teaching certain classes).
There are deals being made behind the scenes. If department Y takes over half your position, what might department X offer them in return?
If you want to bring this about, you'll need to spend some time crafting a compelling argument. There are many directions you could go in, but ultimately it will depend on the department's needs.
As I mentioned above, tenure cases can be really tricky for joint appointments. Both departments end up with a de facto veto, since if one turns you down the other isn't likely to be enthusiastic about coming up with the extra money to save your job (taking the other department's reject doesn't feel good). Before tenure, I'd consider a courtesy appointment to be a much better idea. Once you get tenure, you can then reevaluate and see what your options are.