Yes. The district has strict state and federally-mandated timelines for responding to your request for an initial evaluation, and then for proceeding with the steps to conduct an evaluation, even though your child is enrolled in private school. Child find makes them responsible for every school-age resident of their catchment area. In most states, they have 5-10 school days to respond to your request for an initial evaluation, generally with a request for more information (your concerns), a permission to test, or a refusal to act. But they have to do something. And yes, it has to be on paper.
It sounds like they are within the timelines for responding to you. Once you receive a permission to test (it may have different names, depending on your state, but it will clearly indicate that the district is proposing assessments of your child, should name the areas in which they will assess (e.g., psychological/cognitive, academic achievement, attention/behavior/executive function, social-emotional development), and may even give the names of the assessment instruments), determine if you agree with the assessments, and request, if need be, any additional assessments (e.g., occupational therapy, speech and language, adaptive physical education). For an initial eval, academic achievement is usually comprehensive, but in some cases, you will have to specify assessment in all areas of academics, including reading, writing, and mathematics).
Once you are comfortable that the evaluation will cover the areas of need, sign and return it promptly, as the timeline for completing the testing and holding the eligibility meeting doesn't begin until they receive your signed consent to test. At that point, the district will have anywhere between 30 calendar days and 60 school days to complete the evaluation, depending on the state (check your state's department of ed, special education due process timelines for this info). And then may have an additional 10-30 days to hold the meeting. States may have any combination of, an evaluation review meeting, an eligibility meeting, and an IEP development meeting. Some roll them all into a single meeting, while others split them out, each with their additional time line.
