I have a respirator for work (unpowered but similar to the one pictured by mdcc tester) because I can be working with hazardous and contaminated materials. One thing that we have to do on a regular basis is to under go a face fit test. If it isn't forming a proper seal around the mouth and nose, then it is effectively useless. There isn't a single "pollution mask" for cyclists on the market that would pass a face fit test so for this reason they're fairly pointless right from the off.
We're also advised to take regular breaks while using them in part so that the can be swabbed out of any build up of sweat and moisture from exhalation. Not going to be a lot of point to a pollution mask if you're having to take it off on a regular basis otherwise you'll just be recycling manky air.
As mentioned by Jonny69, the masks just take out the big stuff (unless you don't change the filter very frequently) which is only a small part of the problem. So really, what they're doing is making the air smell better. If that makes you feel better about riding in the city then you absolutely should do it, but there isn't really any compelling reason other than that. In terms of exploring options, you're better off exploring different roads and riding those with less traffic on them and joining campaigns to reduce traffic and place tighter controls on vehicle emissions.
I did use to wear one many years ago and there is one important tangible benefit. When it's properly cold outside, they're a lot more comfortable than a scarf or a buff over your mouth for keeping you warm. I suppose technically they may also offer some defence against Police use of mobile facial recognition as well if that's of concern to you.