Regarding the OP, it depends on when the student goes back to school and how formally the private school documents what the student did there and how well the student does on tests at the beginning of the new school year.
In my state at least, if a student skips a whole grade and changes schools, the school can't force the student to repeat the grade (kindergarten is an exception, I believe). My youngest worked a year ahead of a skip in a tiny private school and then moved to a medium-sized mainstream school. They accepted the skip but made her repeat some of the accelerated subjects.
My eldest homeschooled for a year and I taught him algebra 2. He went to the local high school for year after that. They wouldn't give him credit for algebra 2, but let him enroll in pre-calc. YMMV. He may have got lucky, because the decision maker's hair was on fire over something that day. I don't know. They would have accepted the algebra course had it been taken with an accredited organization.
Dual enrollment: again, it depends. In our district, students in college courses get something like 3 or 3.5 times the number of credits the high school would give for its course. This seems fair; my now dual-enrolled son took introductory chemistry for majors as a semester course, and it's reasonable that he should get way more credits for that semester than someone who covered less material in an entire year of high school chemistry.
I highly recommend dual enrollment. While some community college courses can be gut courses (especially some of the gen. ed. courses), HG+ students who choose their important courses carefully can end up with professors who are very knowledgeable and who can stretch them. My observation via my son: courses taught online aren't likely to stretch; courses where homework is done online (especially math, physics, or humanities) aren't likely to stretch. Asking other motivated students can be a good way to find the best professors. I'm referring to community colleges (well, maybe four-year-colleges too) here, not to gifted-oriented courses like those at AoPS or Stanford's EPGY & online high school.
