Routers route between networks. For them to work correctly you require unambiguous addressing. Duplicate addresses like in your diagram prevent proper operation unless you build workarounds.
The most reasonable thing to do is to renumber one of the networks, so that host and network addresses become unique.
Another option is to hide the remote addresses behind a translation scheme - e.g. you use 192.168.255.0/24 to refer to the right-hand network from the left side, and 192.168.254.0/24 for the opposite direction. The routers in the path then translate the addresses to the ones actually used.
The left 192.168.1.1 would address the right one by using 192.168.255.1 as destination. One of the intermediate routers would translate the destination to 192.168.1.1 and the source to 192.168.254.1 (for the return path). The reply packets would require translation in the reverse way. This option is generally very cumbersome to work with and should be avoided (like NAT in general).
The other problem in your scenario is that the private networks are connected across a public path. If your route packets out to the public with private source addresses they'd either be filtered right away by your ISP or their replies could never find their way back.
To use transparent, private addressing across a public address space you require tunneling: your privately addressed inner packets are encapsulated by publicly addressed outer packets that can be forwarded between the routers. The remote router removes the outer encapsulation, and the private packet can then continue to the destination host. Usually this kind of tunneling is combined with encryption, so you'd have a VPN link.
Another option (that Ron has hinted at) is to hide the private addressing behind a public-to-private translation scheme both ways (aka port forwarding or destination NAT), e.g. the left 192.168.1.1 would use 184.108.40.206:8001 to access 192.168.1.1:80 on the right side, 220.127.116.11:8002 for 192.168.1.2:80 and so on. For more than a small number of exposed services that quickly becomes extremely cumbersome.