I go to the Boat School, not WP, so take that for what you will...
USNA tends to send 6-7 mids on semester exchange to USAFA and USMA, and 2-3 to USCGA, only during the fall. I would assume USMA sends similar numbers if not the same. It is fairly competitive as We don't want to send subpar people to represent USNA at other academies.
In regards to cross commissioning, AROTC-dad is right in that it is not something very common, but there are usually a few each year. I know 2017(at USNA) has one going into the Air Force, and 2016 had one commission into the Coast Guard. It seems to me (based on casual observation, not on hard facts) that there are more people trying to commission into the USN/USMC from other Academies than there are USNA midshipmen trying to commission into the Army/Air Force. I know that there were 7 USAFA cadets and at least one USMA cadet that were trying to commission into the Marine Corps for 2017. I don't know how many of them were actually selected. There are at least 4 USAFA and 1 USMA cadets that are trying to go USMC for the class of 2018 (They have to go to Leatherneck over the summer with USNA mids so this is how I know).
I can't speak specifically to the process because it wasn't something I explored, but it's definitely the sort of thing that you communicate to your chain of command early.
Another thought I would add is that if you are dead set on joining a particular service, then you should put all of your efforts into going into that service. If that means applying to USXA 2 or 3 times, or doing XROTC, or even OCS or enlisting if none of that works out, then so be it. I don't think its good form to go to USMA with the express intent of cross commissioning.