You both should have your own accounts.
Setting the server to
online=false is dangerous and really discouraged. If your server is accessible from Internet, you should never set it that way. So only if you don't have access to internet (like LAN playing somewhere in the middle of forests), you can set it to
online=false, because otherwise you won't be able to play.
Also, changing display name is not enough since 1.7.10 and later. The game uses now UUIDs rather than player names, those are bound to Mojang accounts and are allocated on client login/launch. The launcher sends your credentials and password to the Mojang servers and if correct, servers will send back your UUID and session token. Launcher then can send the request to get the player name with using the UUID. The server then uses UUIDs all over the place, only for few things (those visible) are names used. Server takes the UUID and translates it into names using same method, asking the Mojang servers.
For changing your login name(or playing for 2 names without 2 accounts), you would need to use cracked software, that is again discouraged and dangerous for your computer and credentials safety and won't be helped you with.