Sometimes, immediately after tuning in you'll get sound from both sides, but
one side then fades.

This is an artefact of FM stereo tuning on digital tuners: initially, the tuner
quickly locks in on mono audio, then is able to acquire/decode stereo after a
second or two. So the transmitter is in fact only sending audio on one side,
but this gets temporarily copied over to both speakers when first acquiring the
channel.

(I had a car audio transmitter that was doing this, but only for songs played from
the SD card -- audio through the line in cable was going through on both channels.
So to verify that the transmitter wasn't actually starting off stereo for those first
few seconds, I did this:

- Play one side only through the line cable
- Tune the station up and down

When first tuning to the one-side-only broadcast, sure enough, I got sound
through both speakers.)