My hall bathroom & master bathroom are back to back. When I take a shower In the hall bathroom; water seeps out from under the base the bedroom/bathroom wall at floor level & into the master bedroom. It is not coming out of any of the master bathroom plumbing or shower drains. What could be happening here?

You probably have a leak somewhere between the shower/tub in the offending bathroom and the point where it joins with the pipes from the other bathroom. That sounds like a lot of water to seep under the wall. Sounds like you're going to have to open the wall to find it.

What could be happening here? My guess would be the tile grout. Tike grout will shrink over time and allow the shower stream to get past the tile, where it hits the wall, and run down the wall onto the floor, The solution would be to remove the old grout where the stream hits the wall and regrout. Good luck, Tom

My suggestion would be to first determine whether the water is coming form the hall bath shower drain or form the water supply system. Get a bucket and fill it with water from somewhere other than the shower. Pour it down the drain. Pour about 5 gallons down the drain. Look to see if there is water coming out where you have seen it before.
If so, then the leak is in the drain system. You can then begin looking for the leak in the drain pipes.
If you see no water, stop by a hardware store and purchase a 1/2" PVC threaded pipe cap. Remove the shower head, screw the PVC pipe cap onto the end of the shower arm. Turn on the water and look for water coming out of the wall. If so, you can then begin looking for the leak in the water supply pipes. It will probably be where the shower arm screws into the drop ell in the wall. Pull the escutcheon around the shower arm away from the wall. Using a flashlight look inside for signs of water.

A leak in the water supply system almost has to be where the shower arm screws into the drop ell because anywhere else it would leak all the time, not just when you use the shower.