I love that you aren't blaming anybody. They are just three, and you being a childcare provider, I know you know that three-year-olds are notorious for "MINE!" "MINE!" "MINE!". They don't want to share, plain and simple. Sometimes, it's even unreasonable to expect them to, depending on what it is, and if they are tired, etc.
I think that this is going to take a bit more guidance on your part though. I would definitely tell the other Mom, maybe she can work on her child's sharing skills from her end while you work on your daughter. Also I would maybe participate with them more in play as opposed to letting them work it out. Maybe some 'structured play' where they have to work together as a team, or some activities that will help the two of them get to know each other better.
Give it a few weeks, they will get better. They have to get used to each other. If there's one thing I know about three year olds, they sure are territorial!!