How do I find the path of a file on another computer on my network?
It would depend on how you want to access the file. Here's what I do:


Initiate the connection to the remote computer (I do it via Finder -> Go -> Connect to Server [or Command+K]) but you can also do via terminal using the mount command
Connect to the shared folder/volume available to your user
When connecting to those in Terminal, you would now use the path /Volumes/[name of folder]/[rest_of_path_here] to access them.


For example, if you connected to a folder called Stuff with a subfolder called Things, and ultimately file jackpot.txt on ip 10.10.1.5, first connect via smb to the server (10.10.1.5), choose the folder Stuff, and then reference that path in Terminal as /Volumes/Stuff/Things/jackpot.txt

EDIT: You can also run a Find command (Command + F) inside of Finder while you are connected to the share if you still don't know where the file is. Do the following:


Open Finder, connect to the server/volume (Steps 1+2 above)
Run Command+F to Find, enter your search term
Above the Finder section where files display, switch the search to "Shared" - this will run the search against connected volumes
Locate the file you want, and then drag+drop it into an open Terminal window
The path will be shown in Terminal