Im going to be honest and try not to sounds harsh.
There are many hair, beauty and nail salons that work on a basis of self employed staff renting chairs or rooms etc and work exactly the same hours and it works just fine. Because every client is different and they will choose the salon based on how they like it and from the salon they will choose their favourite therapist to go to.
Seeing girls working in the same salon as 'Direct competition' to me is not a positive attitude to have. Working together, sharing ideas and working on ways to encourage clients into the salon is the first port of call, clients will chose between the two of you based on their treatment experiences.
The 'I need to do treatments no one else does in the place and need hours when no one else is working' just screams a bit of insecurity to me. Basically you want clients who want say a shellac treatment and come at a particular time of day to have no choice but to choose you and not see there is someone else working next door.
In my opinion if you are skilled and likeable, you could work alongside 10 therapists doing the same treatments and clients will always choose to book in with you.
Fear of competition kind of lets slip that you think your clients will wander if given other options.
You have to get some CONFIDENCE in yourself, your products and your service. When you work on this you wont care about competition as you will be turning good amounts of money.
When you think about marketing, think about selling yourself, not about getting clients before the competition can get to them.
Also, make friends. If you are there for a while, there WILL be a time when you need this girl to lend you a hand by lending something that wasnt delivered, or taking a client on a day you are sick that insists that her nails must be done for her wedding etc.
Relax and get some confidence. The rest will follow.