I'd agree with catherine you do need to provide more details.
If you say a public ip as "the one the internet sees" then you have to be specific that have you purchased that public ip as your isp will always nat your ip with its own public ip pool.
If you have two computers with the same isp, they can be at a single location in which case you do not need to involved your isp and set them up in your own home network. They will still have different ips and will be connected to the same isp.
If they are at different locations (Assuming you don't have purchased a public ip for neither of them) then you will have routers at both places connecting them to isp so you can setup a VPN connection between them using a software like team viewer.
If you have purchased public ips for both of them you can just use Remote Desktop Viewer or Run/mstsc in windows to connect to the other computer in windows, if you are running linux you may need to install vnc or rdp clients but you can use ssh without any additional software. just go to terminal type ssh and the ip of your other computer.
I am not trying to insult your intelligence here by giving so many options many of which you would be already aware of, its just that i have no idea of the issue you are facing and what do you have right now. So please elaborate