That does not sound "normal"
When the battery switches are "Off" the only thing that should still have power is your bilge pumps.

Are both engines on/off or is one an off-1-both-2 type?
not sure what you mean by "both positions"

 

Every boat I have had the tilt trim would work with battery switches off as well as bilge pumps
There are two battery switches the one closest to rear of boat says 1, both 2 off
It was on both
The switch towards the front side I think says off and on
The one switch was in the both position
Nothing on starboard motor works
It didn’t power up until I turned the front battery switch to on position

 

Sounds to me like you've got one or more batteries that may need replacing combined with a potentially non-conventional wiring/switching setup.

Batteries are easy to diagnose. Grab a tester, cheap tool, great peace of mind:
https://www.amazon.com/gp/product/B000AMBOI0/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1