Definitely bring it up with the student first to see if you can work the situation out on your own. Sometimes a student will only slack off if they can get away with it unnoticed, so bringing it up to them first will hopefully make a difference. Try assigning them a specific task to see if that helps. Maybe they're not great at taking the initiative but are still willing to participate if you specifically give them a part in the project. While that's not ideal, at least you'll be able to get the work done and everyone will have completed some of it. However, if this doesn't work, then I think you're definitely justified in going to your professor.
Yes!! A lot of people don't want to because they don't want to be seen as a snitch, but it's not fair for you to pick up extra slack and for that person to get a good grade for not doing anything!! Especially at the college/grad level, people need to be pulling their weight. If you feel as though talking to the person directly won't work, it's time to get the teacher involved. Either meet them in office hours and let them know what is going on, or email them. Make sure to document everything you have done and everything the other person was supposed to do but didn't. If you have the proof, the professor will give you the grade you deserve while making sure the other person doesn't reap what you sowed.
Elana,What a great question! This has actually happened to me several times, so I'll answer with experience! :)
The first time this happened to me, I didn't bring it up to the professor. I simply didn't want to be "that girl." (That was definitely me in high school.) But when we all received failing grades, it was hard for me to discuss what had happened with the professor after the fact. He was reluctant to 1) believe me and 2) change my grade because I hadn't come to him beforehand.
The next time this happened to me, the first thing I did was email the TA (teaching assistant) for the course and "CC" my professor. I included GroupMe screenshots and even "BCC'd" them on my group emails about meeting times. Even though the professor low-key mentioned group collaboration in class, my group continued to completely suck.
When our project was barely satisfactory, it was a lot easier for me to make my case. I didn't even have to bring it up to my group -- they don't have access to my grades. So while they all made C's, I made an "A," because I could prove that I did the work.
I hope this helps you! :)