OK, I have figured out that a 60 watt incandescent bulb puts out about 800-1500 lumens, and a 20 watt LED about 1000-2000 lumens, so I guess the equivalency is correct.

That said, how many lumens should I be looking for for each side of the vanity mirror, and what kind of bulbs?

that depends on what type of light you like. We have really tended away from "soft white" light recently---it reads very very yellow to us. We have been using more of a "bright white" light. Which for us means some of the 3000K LEDs--but not all--some look too yellow and some look to blue....we have also used GE reveals for a whiter light. In our recent master bath redo--it's a dark room, shaded backyard--very little natural light. I put in 2 recessed lights with 50W halogen floods (whiter light than incandescents) we well as a 150W (3 50W) halogen fixture over the vanity, and the shower has a light as well. it's an 8x8 bathroom.

Contractor felt the overhead lighting was overkill, I said I'd rather have the light and not use it than need it and not have it. so, we have it, we need it and we use it!!

I am not sure it helps, but we went with whiter light rather than yellow light.