I tried using the mic that came with the PS4, but I didn't like the way that it fit in my ear. Instead, I opted to buy a pair of earbud headphones that included a mic. I plugged it into my DS4, but I only hear audio out of the left earbud. I would be okay with this, except the mic on this pair of headphones is attached to the wire to the right earbud. Is there any way to get the PS4 to push VOIP chat through the opposite or both headphones?
First, ensure that your headphone plug is pushed all the way in. Hearing only one channel of audio and the other earbud being a (poor) microphone is exactly what one would expect if the connector is plugged in most, but not all, the way.
There are various standards for the connector for combined headphone/mic systems. The PS4 uses a TRRS jack, so your plug should have four metal bands (for left, right, ground and microphone) separated by three plastic insulating bands. If you don't have all four, that's your problem, though, if you have three, stereo audio output should work correctly (again, if it's pushed all the way in).
If you do have all four you've run into a problem of differing standards and your left, right, ground and mic signals are not in that order, which is the one expected by the PS4 controller. You may be able to find an adapter, but if your connector really is pushed all the way in, your current one adheres to neither of the two most common standards and you may be best off just buying a different headset.