I always play animations on the client, and they replicate just fine to the server. However, I have had some issues similar to this. If you edit any properties of the animation (such as priority or looped), then it will not replicate to the server. Make sure you set the properties as needed in the Studio animation editor before you upload it.
Do you set any properties on the client? Maybe the priority isn’t high enough on the server to override the default animations.
Check this thread also for more info: Where to play an animation, client or server?
EDIT: Above reply has a good point. Try to do a 2 player test and see if the 2nd player sees the animation correctly.