I did both with about equal levels of success. Of course, it depends on how you define "success." When you're at the exchanging number phase, I define success as a follow-up contact that leads to an in person meeting. From there all the usual rules apply. If you don't have chemistry, you don't have it. But you don't know that before giving your number or trying to get hers, so it's worth the shot.
If found giving my number to be less threatening in a world of growing uncertainties. I found asking if I could call to be assertive, but can be done in a non-threatening manner. And as I said, I didn't find one method to be any more successful than the other. I had good results and bad with both.
Perhaps the best example of me giving my number was to a woman who lived in the same building as I did. It was a mid-rise just outside downtown Denver. She was two floors above me and we met on the elevator one morning. In the few minutes it took to get to the ground floor and out to the parking lot, we really hit it off, making overly polite small talk and showing more interest in each other than most I'd meet in the elevator.
When we got to the parking lot she went her way and I went mine. I kicked myself for not trying to get her number or give mine so I quickly started scanning the parking lot for a moving vehicle. I spotted her car and made a mental note of the make and model. That night I walked up and down the parking lot until I found her car and left her a note.
The next night she called me. The night after that we went on a dinner date. We followed that up with a show the next night. By night 3 she was riding me like a bucking bronco. I had no shot of getting her off of me, though. This is the same woman I've mentioned before, who was 6'2" and very plus sized. She was a beauty. A light-skinned AA woman of 37. I was 23 at the time.
She had a big butt, thick legs, trunk, and boobs so big I almost didn't know what to do with them! We went out in public and she'd wear heals, which made her tower over my 5'7" self. And she had an appetite that had clearly not been addressed for a while because she wore me out! Good thing I was young and pretty much indestructible back then, though I did hurt my back with her one night.
Sadly this is also the same girl who raised a political discussion, said her piece and tried to shush me when I tried to counter with a point of my own. That was the start of us splitting up. It was fun while it lasted though!