H.W.
Hi W.,
Children at this age don't really understand 'sharing'. I teach a group of three year olds who actively struggle with the concept, so don't worry.:) What I usually recommend is giving your little girl plenty of space to explore the toy she's playing with; when she goes after another child's toy, tell her "Oh, I see Suzy's using that right now, here, you can play with this" and have something as similar as possible in hand to offer to her immediately.
At this age, it's also fine to do a lot of modeling. If she does hit another child, do your best to give that child your attention. "Are you okay?" and then, if it's appropriate, either hold that child (if their parent isn't available) or involve your daughter in making amends. This might mean getting a cold pack, and notice aloud what's happened. "Do you see Suzy's face? She looks very sad. When you hit Suzy, it hurt her." and so on. If your daughter is having a difficult time with other children using desired toys, consider moving your daughter. If she's getting upset because she doesn't want other kids to take her toys, consider letting her play in a pack-and-play where they can't take her toys.
Parenting youngsters takes a lot of intuition and ingenuity in creating safe places/ situations for everyone. As for your being unsure about your role as a mom, let me make it clear that, despite a lot of what's said about how kids at this age can "learn the rules", warnings generally do not work. So the first time the hitting happens, you have to take action. Time out and wholesale removal is less instructive than sitting with her and giving her lots of support. Even three year olds+ have challenges with self-regulation and self-control, so being very present when she is with other children (or giving her a safe place to play on her own where she's happy) is your best bet.
And you're right...she doesn't 'listen' (most early toddlers don't!) , so your actions are going to be more powerful than words. Keep on giving her consolation and empathy when she needs it, and show this to her when other children are hurt (not just when she hurts them, either.) . Your consistently teaching her that everyone's feelings are important will pay off in the long run.
This too will pass--- it just takes them a LOT of time to learn how to share. I know lots of older kids (even 10 years plus) who struggle with it. Remember that she's learning so much right now! Good luck! Sounds like you are already doing a great job of being in touch with your little girl!