In today's age, I don't think it's a problem at all ! Just write your name and number on a piece of paper and ask the person if they would like to have your number. If the person says Yes, then give it to them and maybe you'll have a great date in the future. You never know! Sure doesn't hurt to ask!
What are your thoughts on a girl giving a bartender/waiter her number? How would you go about doing it?
Go for it!!! I believe in taking chances. I've known people who put their number on their bill. Or asked another server to deliver the number. Because a lot of times they are so busy serving customers you have to be inventive or enlist the help of someone else. Good luck!!
If you are in the habit of giving your number to strangers you would write it down, and hand it to them. Not sure what their chosen profession has to do with it.
Just keep in mind they likely get many many numbers from people.