On Thu, 10 Mar 2016 09:40:32 -0500, 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?
A GFCI has 5 connections, ground, line in, line out, neutral in, and
neutral out.

To find which one is first in line check to see which one has more
wires in the box. The one that feeds through WILL have 2 cables
entering the box. The second one may or may not. If both have wires
in and out, disconnect the wires in one box and see if the other is
still "live". If so, repeat at the other box.