Thanks for the response, batti! I did a little more debugging today and it actually does look like I need a bypass capacitor for the receiver. It also seems that the less soldering I do, the better. For some reason, data doesn't travel well over soldered connections I made (probably because they were poorly soldered? Not sure). I wired the same circuit I had done on the perfboard onto a breadboard with jumper wires (plus the extra bypass capacitor), and it seems to be working fine now.
Sorry for bothering everyone! Turned out to be a circuit issue rather than a Tinkerforge-specific issue.