Private Data Beacons aren't scanned with the Data Link Scanner.
All you need is the basic Scan, which will easily work from a few km away.
Just point at it and select the Beacon as Target (either manually i.e. "Target ahead" while pointing at it or via the Contacts Panel). Like you would Target and do a basic scan on any normal Ship.
After a few seconds, the basic/automatic scan will complete and you'll get the various Data it offers.
Note : this only works once, so after getting the Data and i.e. trying this a 2nd time on the same Beacon wouldn't do a thing anymore.