Can a self employed invoice a private person?

Sure. I hired a painter two weeks ago - he is his own company, so is self-employed. He gave me a quote for the work, I agreed to it, he performed the work and gave me an invoice for the work, which I paid.

Absolutely. I invoice private clients all of the time.

Be careful. It sounds like you are just starting in being self employed. I would get paid on the spot for a while. If you don't have a printer, get one. Or buy some blank invoice forms at Staples and fill them out by hand, more as a receipt than an invoice. Some people are just stalling, or will never pay. A job for a private individual doesn't need to be billed. Don't be too polite to say pay now and here's your invoice.

Joy's advice makes sense.

I have a small handyman business as a retired "something to do" job. I do limited work for seniors who are referred to me. I get paid at the end of the job and always give a receipt for work done.