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Well, as Jon has already noted, packet is forwarded only by its destination IP. Which means, if the original packet has only a private IP in its source address, it won't be involved in the forwarding decision. (Again, the source IP might still be subjected to analysis that will block the packet at some point.)
So, your private IP will only be a routing consideration if the receiving host is trying to reply and use your original source IP as the return packet's destinations IP. (I suspect you understand the foregoing, but I did want to insure there's no misunderstanding.)
Ok, so if the original destination host generates a packet with a private IP for destination IP, it would be (most likely be) treated, from that host, like any other packet that host generates with a private IP in the destination. I.e. The network will attempt to deliver it.
If the prefix destination is totally unknown, the packet will be dropped unless the forwarding router has a default route (or aggregate) that covers it.
Assuming there's a local private subnet, that matches the destination, the network will deliver it to that network, and if there's a host with that actual private IP, the packet will be delivered to it. Usually, the overlapping private IP host will drop the packet, as it won't have process expecting the packet, but it's possible, a process is might accept the packet and attempt to process it. Then, most likely, the process will go "huh?" and drop the packet. However, it's also possible the newly receiving host will reply to your original receiving host, i.e. those hosts will now fling packets back-and-forth, because of your original packet. Again, this is all very unlikely normally, more so if the network isn't "sloppy", but such routing is the basis for some DDoS attacks. (For example, I place another host's IP in my packet's source IP, and then send out ping to the network broadcast IP. Hosts receiving the ping will send a ping reply to the host I targeted.)
