I can't imagine why you might need this since for the remote instance's full path is \danas-pc\PublicDoc but if you let your imagination thrive I'd suggest something like this:
(1) on the remote computer inside the share folder you can drop a small script that if executed return the full path. You have to search for appropriate coding for windows or linux environment also you need to have execution privilege or rights on it. for example on windows you can have a vbscrit or cscript and a .sh script in linux.
Also please note that seeing it from the remote host, in terms of the remote host the full path is \NAME-OR-IP\Path\to\Folder\or\File etc. For you on the remote connection that is the full path ;)
UPDATE:
as per the comment below, this is a full script that does the following
- creates a vbscript with code in it to retrieve the current full path
- copies the files into the network desired path
- executes the vbscript and reads the result back
- deletes the vbscript
Assuming: you have the read/write access on the network folder
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Net.Mime;
using System.Runtime.CompilerServices;
using System.Text;
using System.Threading.Tasks;
namespace GetNetworkFullPath
{
class Program
{
static void Main(string[] args)
{
var networkFolder = "\\\\REMOTE-PC-NAME\\SharedFolder";
var nameOfVBScript = "capturepath.vbs";
var vbsOutput = "";
//Get the name of the current directory
var currentDirectory = Directory.GetCurrentDirectory();
Console.WriteLine("Current Dir: " + currentDirectory);
//1. CREATE A VBSCRIPT TO OUTPUT THE PATH WHERE IT IS PRESENT
//Ref. https://stackoverflow.com/questions/2129327/how-to-get-the-fully-qualified-path-for-a-file-in-vbscript
var vbscriptToExecute = "Dim folderName \n" +
"folderName = \"\" \n" +
"Dim fso \n" +
"Set fso = CreateObject(\"Scripting.FileSystemObject\") \n" +
"Dim fullpath \n" +
"fullpath = fso.GetAbsolutePathName(folderName) \n" +
"WScript.Echo fullpath \n";
//Write that script into a file into the current directory
System.IO.File.WriteAllText(@""+ nameOfVBScript + "", vbscriptToExecute);
//2. COPY THE CREATED SCRIPT INTO THE NETWORK PATH
//Ref. https://learn.microsoft.com/en-us/dotnet/csharp/programming-guide/file-system/how-to-copy-delete-and-move-files-and-folders
string sourceFile = System.IO.Path.Combine(currentDirectory, nameOfVBScript);
string destFile = System.IO.Path.Combine(networkFolder, nameOfVBScript);
System.IO.File.Copy(sourceFile, destFile, true);
//3. EXECUTE THAT SCRIPT AND READ THE OUTPUT
//Ref. https://stackoverflow.com/questions/27050195/how-do-i-get-the-output-from-my-vbscript-console-using-c
Process scriptProc = new Process();
ProcessStartInfo info = new ProcessStartInfo();
info.WorkingDirectory = @"" + networkFolder + "";
info.FileName = "Cscript.exe";
info.Arguments = nameOfVBScript;
info.RedirectStandardError = true;
info.RedirectStandardInput = true;
info.RedirectStandardOutput = true;
info.UseShellExecute = false;
info.WindowStyle = ProcessWindowStyle.Hidden;
scriptProc.StartInfo = info;
scriptProc.Start();
scriptProc.WaitForExit();
bool exit = false;
while (!scriptProc.StandardOutput.EndOfStream)
{
vbsOutput = scriptProc.StandardOutput.ReadLine();
}
Console.WriteLine("vbscript says: " + vbsOutput);
//4. DELETE THE FILE YOU JUST COPIED THERE
System.IO.File.Delete(@"" + networkFolder + "\\" + nameOfVBScript);
}
}
}
Unfortunately when executed remotely the script replies with the Network Path :( so disappointed...really sorry! As long as execution is happening from a user outside the remote system it will reply with the absolute path related to that instance. I think an internal process/user should execute the file and reply back with the answer to the application.

I'll try to think something more tomorrow and maybe reply back if I'm lucky.