Anyway, do you think this situation is actually toxic? Is eating from around the driveway just a bad idea regardless of the specifics? That's what I was really trying to determine. I am hesitant to waste such a gift, but of course I want to avoid actual harm.
The only way to know would be to test it. It's going to depend on how often a car is stopped and started there, what kind of fuel it uses etc. You can get pollution from exhaust emissions, fine particles from tyres (including heavy metals), anything that is used to wash the car, accrued road pollution that washes off the car in rain etc.
Toxicity of anything is related to dose. If the drive and soil underneath are heavily contaminated you could eat the purslane once and be fine, or eat it daily and start accumulating toxins. It's also related to your overall toxin load and your body's ability to process that.
I tend to agree with the comment about the lettuce and I have harvested wild greens from roadsides, but the idea of harvesting out of a drive like that doesn't sit quite right with me personally. On the other hand, if I wasn't getting any other high nutrient greens I'd probably eat the purslane while I propagated some elsewhere.