This was tested in multiplayer MCPE to MCPE and MCPE to Minecraft Windows 10 Edition. Both produce the same result.

Steps to duplicate
-Host a local server.
-Have the client connect.
-Have the client place 3 blocks of nether rack on the ground in front of the host, then light them with flint and steel.
-Have the client put out the fire.

The fire on the clients end will go out. The fire on the hosts end will keep burning as if nothing happened. This also happens when you light the grass on fire as the client then put it out. The fire will keep burning on the hosts end.