In the science and engineering fields in the US, where nearly all graduate students are supported by teaching or research assistantships (TA/RA), academic departments have to decide whether an applicant is adequately prepared to be a successful student in the program and whether the student can do the TA/RA job. When the student comes with their own funding, that second question doesn't have to be answered.
In my department (mathematics) it's quite common to have an applicant whose academic preparation is good but whose English isn't good enough that the student could be successful as a TA. We would typically admit the student without offering the student an assistantship. Most applicants that are offered admission without an assistantship don't enter the program. However, if the same applicant came to us with their own funding, we'd admit the student. So, having your own funding might help.
There are some costs associated with having a student even if the student has their own funding for tuition and living expenses. Supervising a graduate student takes faculty time, and some faculty are only interested in supervising PhD students and less willing to work with MS students.
Also in the lab sciences and engineering students need access to research facilities, equipment, and supplies to conduct their research. Students who are funded as RA's under a grant also come with grant funding to cover these costs. Depending on your area of study, you might have to convince the department (or perhaps an individual faculty member) that you would be able to contribute enough to the research activity to justify these expenses.