The Animation Blueprint in the Third Person template works by ‘inspecting’ the Character each frame for it’s speed/jumping state, and updates the animations accordingly. Our Character class automatically supports networking, so it will move on client and server, and hence the animation will work as well. If you are not using our Character, or you are adding functionality beyond basic movement, you will need to send the network messages yourself to get the animation to play. You can check out these videos on how to get networking set up for Blueprints: Introduction to Blueprint Networking - YouTube
I hope that helps!