Pysgod has it right for the merchants. You will need the intimidate perk, and the ones I've found that allow you to intimidate them are usually stopped temporarily along side of the road. Usually with a horse.

Beggars I haven't gotten anything towards the achievement for yet. Someone in a thread I created stated they have found three that they were allowed to give gold to in each town, and after giving gold to 5 they got credit for that area towards the achievement. So far I've only found two beggars in Daggerfall, and then a lot that ask for gold but don't actually have an option to interact. It looks like beggars may be the same as the merchants where only specific ones give the interact options for the achievement.

I've found 2 beggars so far, one in the wilds, and one in the cities (daggerfall). I've found a few merchants but haven't bought into persuasion yet.