It looks like your putty sessions are stored in the registry.
Start up the registry editor: regedit.exe (on Windows 7, hit the windows key and type regedit.exe)
Browse to HKEY_CURRENT_USER\Software\SimonTatham
Right click on "Sessions", choose export. Give a name. Email the file.
On the other computer, get the file, save it, right click on it, click on merge. Done