I've never let her have it before because I don't want her to choke on it, but when I took her to her doctor's appointment, the sheet of information they gave me mentioned gum and hard candies were okay with supervision, so I decided to let her try a piece as a special big girl treat for being 4 years old.
We had to go to the pharmacy on the way home and I told her she could pick out any pack of gum she wanted and chew her first piece when we got home. She was so excited and very well behaved while I got what I needed at the store (amazing how far a little motivation goes). Of all the different kinds of gum, she picked the same strawberry trident that I chew all the time, carried it to the cashier and told her "that's my gum - I get to have gum now because I'm 4."
With her piece unwrapped, ready to try it

She was so excited I had to capture it on video...
She loved it and I have given her several pieces since then - she is very diligent about wrapping it up when she's done and holds the empty wrapper the whole time so she can throw it out in there when she's done, and then of course hands me the wrapper filled with gum. She swallowed one piece by accident and was very concerned about what was going to happen since she knew she wasn't supposed to swallow it. She'll ask questions like "is it okay if I have gum in my mouth and wash my hands at the same time?", it's just so cute how excited she is about it.
Growing up, Mom always had gum in her purse and I remember saying one time to her how nice it was that she was always willing to share her gum and that I couldn't imagine ever wanting to share my gum as freely as she did. I am proud to say that instead of selfishly hoarding my gum as I envisioned I would, I am carrying on the mother-daughter gum sharing tradition :).