No - you cannot send objects around. As marc_s pointed out, WCF is a message-oriented communications framework.
But, you do have different instancing options.
By default, Windows Communication Foundation instantiates services on a per-call basis: a service instance, a common language runtime (CLR) object, exists only while a client call is in progress. Every client request gets a new dedicated service instance. This is something like Stateless Session Beans in J2EE.
Another option is session-based activation, which is something like Stateful Session Beans in J2EE. Using this approach, when the client creates a new proxy to a service configured as session-aware, WCF activates a new service instance and attaches it to the session. Every message sent by the client over that proxy will go to the same instance on the server side.
This activation behavior is selectable with the ServiceContract
attribute.
Juval Lowy has written a good article on instantiation options in WCF.
Within these instancing options, you may find something that works for you.