I have 10 and 6 year old children and I slept with them untill about they were 3,4. I show them many books and tv shows which includes a scene of that parents say good night to their kid and close the door, all the time. Ocasionally I said "you are big enough to do this, too."
It's much easier not to sleep with them at all. But once you did, you need a lot of pacient.
I am from Japan. Japanese children tend to sleep more nights with parents I think. We don't worried so much about children. They grow out of it naturally if you tell them and just wait, but sure it takes many years, 5 to 10 maybe. Every children are different. Here in the States, you can't see those differences because children sleep alone since they were born. It's about parent's time. It's heard for them to stop sleeping with mom and dad once they started, I know. But It's about parent's time.
I really don't like the idea to let her scream for a few nights because she is already 3. Little infants can fall in sleep after craying, but 3 year old children are tougher than you think. It's much hearder than you think. I know one mom tried. My friend daughter didn't fall in sleep for 3 hours, screaming, banging her door, etc ... for 5 days. She slept on a floor near the door. Her face was all about craying, you can see. The little girl lost her voice after couple of days. Day time was awful, no happy time, always scared of that another night is coming for both mom and daughter. I can't explain well enough that that was awful. The mom, my friend told me she had to stop it before her daughter start to hate her, she felt like. They had a last talk. The mom told the little one, "I will sleep with you 5 nights and you sleep one night in your bed. And in our future, I want you to sleep in your bed a little more often than that." The little one asked her mom to stay in her room till she fall in sleep the one night out of 5. They did probably for about 6 months. Then it became all right. The mom told me, she just should have had to tell how serious she was and waited for her daughter to be ready, she felt after all.
Good luck.