We’ll use A to represent the first cowboy and B for the second cowboy.
Starting out.. the cowboy A has 1 cow and cowboy B has 3 cows.
A + 1 = 2A, so A = 1.
A + 1 = B – 1, so B = 3.

When cowboy B says “that’s fine by me”, ownership of one of cowboy B’s cows is transferred to cowboy A. The important words here are “double HIS herd”. If ownership of one cow didn’t transfer from cowboy B to cowboy A, cowboy A’s herd (HIS herd) would not be doubled.

Therefore… when the question is asked “How many cows does each cowboy own?”, cowboy A and cowboy B both own 2 cows. Note: Until the gifted cow is physically transferred from cowboy B’s corral to cowboy A’s corral, cowboy A’s corral will have one cow in it (owned by cowboy A) and cowboy B’s corral will have three cows in it (two owned by cowboy B and one owned by cowboy A).