Many of those tester pens are going to read hot on both wires in that situation, but only one is really hot.

If this is the case, I may just pull my hair out. Guess I'll install the fan and see how it goes. Why would it read hot on both though if both aren't hot?

https://en.m.wikipedia.org/wiki/Electromagnetic_induction

The second wire and your tester get a magnetic force the same way. It is what makes your pen light up and also what gives your pen a false reading on the second wire, which has also recieved the magnetic force. Basically, the pen is too sensitive.

Non-contact tester can give false positives under the right circumstances, which, I guess, is better than false negatives. ;-) Digital multimeters will also detect "phantom" voltages but those usually show up with readings in the 50-70 volt range, which is a clue that something is amiss. Like any tool, each has it's advantages and disadvantages.

If you're going to do a lot of electrical work, it might be useful to invest in a solenoid type tester. Your local home center will have them for about $30.

Non-contact tester can give false positives under the right circumstances, which, I guess, is better than false negatives

Not when the tester gets defective and is completely unreliable. I have had a GB tester that wore out after a year, it showed false positive just holding it in the free air. Get the klein or milwaukee tester, you have to hold it against or very close to the specific wire for it to detect live.

The only non-contact testers I've had are from Fluke. The first one was damaged by a battery leak and the second I've had for quite a while now. I can't say anything bad about them.