On Thursday, March 10, 2016 at 9:40:36 AM UTC-5, Wade Garrett wrote:
1979-built house, two bathrooms sharing a common wall, one plain
electrical outlet in each bathroom. Both outlets are on the same
circuit; i.e., one circuit breaker kills them both.

I want to install GFCI protection. Do I need to install one in each
bathroom?

Or is the downstream one protected by its upstream brother? If yes, how
do I determine which one is first in the circuit?

--
Wise men speak because they have something to say; fools because they
have to say something.
�ƒ Plato

Outlets connected to the downstream side are protected by the one
GFCI. To figure it out, if you can get an idea of how the wiring
was probably run, that's a start. If you can't see how it might
be run, I'd just start with the one closest to the panel. Pull
it out of the wall. If it's feeding to elsewhere, separate it,
turn the power back on, see which half is live, which dead and
proceed from there.