Basic water conditioners are ones like API tap water conditioner... usually based on sodium thiosulfate and sodium EDTA, they neutralize chlorine by reduction to chloride, and reduce metals to EDTA chelates..
These basic products have no affect at all on ammonia or nitrite like Prime and Amquel do. These products could be considered by some to be more advanced than just a basic dechlorinator.
Regardless, neither kind in any kind of intended reaction will reduce oxygen.. but all dechlorinators work by being a reducing agent. *any* reducing agent can and will react with oxygen to some degree. Oxidizers and reducers react, and oxygen is an oxidizer (go figure).
Thus, oxygen depletion is a possibility with anything that can turn chlorine into chloride. It's nothing fancy about prime that does this, it's the reducing action of the dechlorinator part itself.
But to be honest, significant reaction with oxygen in any dechlorinator really isn't typical and the doses are small enough that the amount of oxygen scavenged should be modest.
Though the potential to reduce oxygen is there with any reducing agent, this is very, very rare. Prime can be safely overdosed up to five times the recommended amount in an emergency situation. Doing so is very safe, and it would take a massive overdose to have any effect on the oxygen levels in the tank.
If your O2 levels are on the hairy edge, dechlorinators will adversely affect your tank, but perhaps you need to fix your O2 level problem, not worry about your dechlor.
Normal saturation level at 78F or lower is over 8ppm of O2 (cooler temperatures will have higher O2 levels). From what I'm reading, fish show stress below 7-5ppm depending on health, speicies etc...
This site claims typical dechlorinators have the potential to reduce oxygen by 0.5ppm for 15 minutes. That still gives you 0.5 ppm of wiggle room before any stress is shown, and even that is the most sensitive fish, at warm temperatures: