There ARE legitimate reason why this situation will exist (such as school district policies for teams engaging in out-of-state travel). The game manual does clearly address everything from the VEX competition level on what the students can and cannot do.
First, G4 and G5 apply the same whether the two teams are in the same org or not. For example, a senior member of a private org can mentor a freshman member of the school team in general ways - teaching them how to write PID, sharing build techniques for reducing robot weight, walking them through general troubleshooting processes - but they cannot share specific robot mechanism designs, copy/paste code, or provide direct answers to “why” something on a build didn’t work.
To note, this are exactly the same rules for two teams in the same organization. Two teams can share ideas, but cannot co-design or co-build their robots.
A few things not addressed in the rules (but that would fit under the G3 Common Sense rule), are…
First, I would advise that students from the private org should avoid being in an official leadership position in the student club - such as a club President or Treasurer office, for example. This could easily create a perceived conflict of interest.
Second, be VERY careful on ownership of parts. The private teams should not be building their robots in the club space. Nothing wrong bringing their bots to club days to do scrimmages, but having a clear separation of materials is very important. Nothing will create problems in a relationship faster than money. Even the insinuation that a private org is benefitting from the resources of a school club will create… issues.
Finally, the students (and coaches) in the private org should always act in a way that is positive towards the school. This is just an aspect of G1, but definitely worth calling out.
This is all shared from experience. I coach two private teams (private so that they can compete in out-of-state Signature Events - this was our superintendent’s idea). The students on these teams all are still associated with their school club, and all of the above items are my expectations for the two teams. It has been a very healthy relationship, giving the two teams opportunities they wouldn’t have had, and not doing so at the expense of the school team. (Fully self-funded does cause a pain in the wallet, but that’s been our only real downside.)