Dealing with rigidbody collisions when sending player positions across network

I'm creating my first networked game in unity. It's a simple 2D version of "Tag." I've got the game running fine on two mac$$anonymous$$nes connected to each other.

I'm using two methods for sending messages between the mac$$anonymous$$nes.

1) Input-based messaging (sending key codes).

2) State-based messaging (sending positions)

I can swap between both methods.

When receiving input-based messages, the client adds a force in the direction of the input to the remote player's game object, as follows:

By applying force to the object, collisions are handled automatically by the rigid bodies. T$$anonymous$$s results in accurate collisions for both mac$$anonymous$$nes.

When receiving position-based messages, there is no force and thus no "proper" collisions. When players are moving fast enough, they simply pass through each other. It's not possible to have both mac$$anonymous$$nes sending both player's positions across the network. T$$anonymous$$s will cause conflicts.

So I guess the question is, can I handle such collisions without an authoritative server and without sending direction/force vector data across the network (i.e with simple positions)?