Sounds less than a Mirror related problem that just scripting. You could use a dual camera setup for each player, like FPSs usually do to draw weapons on the screen independently of environment geometry (so that the model doesn't clip through meshes if you get close to a wall).

You make two cameras, with the exact same behavior, except that one has its culling mask set to render objects on the current player's client layer (which you have to set in projects props), and the other renders everything except the client layer.

When you pick up an object, you just have to change the layer on which the gameobject and its children are on: you put them on all on the client mask. This way, other player's cameras are simply not rendering it.

This has the downside that, for each players, there has to be an available layer for its Client camera's culling mask to set on (and for the others to exclude). So if your game is 8 players, there has to be 8 layers. It makes this solution rather inelegant, because you rely on the editor max layer count, but except if you plan on making your game playable by more player than you can declare layers on, which shouldn't be the case, then you're good.

To summarize:

• Create as many layers as you can have max player. ie, 8 players game, 8 layers named ClienLayer[1-8]
• Make it so that the player prefab contains 2 cameras, both behaving the same way.
• One camera should have a culling mask containing everything EXCEPT ANY ClienLayer
• The other camera should be given, when connected, a culling mask containing ONLY the ClienLayer corresponding to the player
• When a player picks an object up, you set that object renderers gameobject's layer the ClienLayer corresponding to the player who picked it up.

The result is:

• Each player have one camera that draws the environment only
• Each player have one camera that draws object on a layer exclusive to the Client