From my experience, you need to make sure you have matching database entries on PatGuard 2 for site and location as you do on the tester. You can enter these manually, or I believe you can download the configuration data (sites, locations) from the PT350 using the Send/Receive Config Data option, but I'm not sure how that is downloaded into PATGuard2.
To download records you go to Tools, Download on PatGuard2 (I was trying this out on the Elite trial), make sure it has selected your PAT tester and the correct COM port (if it's the wrong COM port you won't get anything at all downloaded, there may be several listed so keep trying them until you find the one that works; most of them will probably be virtual ports, one will be for Bluetooth serial emulation), select client and site if necessary, then click OK, it should show a box saying Start Transmission. Now on the tester go to Menu, Data Transfer, Download to PC, set the format as PATGuard SSS, then choose Send. A progress bar should display on the tester and a bytes readout on PATGuard, when the transfer is finished successfully these both disappear along with the download window on PATGuard. The records should now be in your PATGuard database.
I learned this just from playing around with the PATGuard2 Elite trial and my PT350. One thing you should check is the settings for the COM port - I found that it needed to be set to the following: bits per second - 57600, data bits - 8, parity - none, stop bits - 2.