The answer to this question is slightly tricky. Many (or most) private schools are not legally obligated to provide such services. If your child has been placed in a private school setting via the public school system's IEP process, then the onus of financial responsibility for providing Special Education services is on the public school system. If a child is placed in a private school by a parent, there is no such responsibility.
 
However, by law, public schools must also have a private schools service plan that provides some services to students with disabilities who attend private school. The funding, services and delivery of services are sometimes limited in scope, depending on the district in which your child would be attending school should s/he attend public school.
 
You mentioned that your child has a service plan. Is this an educational service plan (an IEP or IFSP)? If an educational service plan has been implemented, then the agency who worked with you to create the plan, typically the public school system, is obligated to provide the services outlined in the plan.

The private school responded to my request that they test her, and the district school system paid for that - hence the Service Plan. I know that the district will provide some after school "services" - for which I'm bracing for a struggle as I'm not interested in their tutoring her, but rather giving her learning strategies. I'm very concerned about accommodations in school - other than extra time on tests. I have discovered that it's on me to try to convince them to allow her, for example, to receive note-taking assistance (either via a Pulse Pen which is my choice since it puts the responsibility on my daughter, or a note buddy - which is not a great choice since it's a small school and that could open the door for relentless nastiness on the part of classmates).