What you describe is roaming: a packet meant for network B gets picked up by a gateway of network A and then gets forwarded to B on the network level. The LoRaWAN association is working on this. While the technical aspects are one thing, the biggest task will be the organisatorial part. Making roaming agreements between networks, implementing, monitoring and enforcing mutual fair access policies etc.
Btw: what makes you think that the non-TTN packet you pick up on your gateway doesn't also get picked up by a gateway from the network it is meant for? In this case you would forward redundant packets. If you receive lots of packets meant for other networks this probably means that other network has quite good coverage in your area.