It would not work very well, as far as I can see. Proxying (as well as SNAT/masquerading) requires much more than simple IP forwarding.
Let's see, say that
- 192.168.1.1 is your network gateway
- 192.168.1.2 is Alice
192.168.1.3 is Eve
192.168.2.2 is Bob.
Eve sets Alice as her gateway, and pings Bob. The packet arrives to Alice who then forwards it to the real gateway, whence it would arrive to Bob marked as Eve's. Bob would answer and the reply would be sent directly to Eve, bypassing Alice.
So Bob will never know what happened, and will hold Eve responsible for what Eve did.
The only thing is that now Alice gets to eavesdrop one side of the conversation.
So now Eve forges a packet as coming from Alice. Alice receives a packet from her own address coming in from the outside, and in all probability she drops it.
Eve could forge a packet belonging to Charlie, but then, she could do this even before, without Alice's forwarding support. Actually, she could forge anyone in the local network; by setting Alice as the gateway, she is now worse off, for she can forge anyone except Alice.
If she has some way of foreseeing Bob's responses, she can engage in a blind spoofing of the conversation (but again, she could do this before), if she can make it so that Charlie does not respond by aborting the connection, and that the gateway doesn't inform Bob that Charlie's not reachable. Also, for connection protocols such as TCP, "foreseeing Bob's responses" is not all that easy unless Bob's TCP stack is vulnerable to prediction.
If the real gateway has security on it, now she can make it look as if Alice was forging a packet from Charlie, so that the real gateway spots the forgery and believes Alice to be responsible.
This last seems the most feasible "attack" that Alice exposes herself to by providing forwarding services.