It won't work. There is no way to send a packet to another machine with the same address (192.168.1.10) The traffic will be routed directly to the loopback address and won't even leave the system. Your diagram shows an addressing conflict. Personally, I would change the address on the right to a different address.
You could use DNAT to make 192.168.1.0 on the left appear as a different address and route to it. This is done when bridging private networks from different organizations. The appropriate DNS overrides on the local DNS servers are required so that the servers can be reached by name.
EDIT: You need a router where you can configure DNAT individually by interface. Linux based routers should work, but I don't think dedicated routers have this capability. Typically, NAT is handled by firewall devices. Linux routers generally combine routing and firewall capabilities.
EDIT2: When you are configuring a network which is likely to be connected to others internal networks it is best to use less popular sub-nets of the 192.168.0.0/16 block, or a block in one of the alternate private networks, 10.0.0.0/8, or 172.16.0.0/12. Pick a random number between 4 and 255 for your sub-net (second octet for 10.0.0.0/8, third octet for the others). If you use the 172.16.0.0/12 range pick a random value between 17 and 31 for the second octet as well.
If you distribute addresses using DHCP, you should be able to change the sub-net relatively easily. You may want to route the old sub-net while the addresses are changing over.