EET does not require you to give a receipt
The word "účtenka" does not appear in the sales recording act. There is no duty to issue one and no penalty for not doing so. Almost everything written about a compulsory receipt describes the old EET of 2016, which was a different system.
01
You may still have to issue a document — under the VAT act
If you are registered for DPH, §30a requires you to issue a simplified tax document, and §29(1) gives the customer the right to demand a full one. That has nothing to do with sales recording and applies regardless of it.
Based on: zákon o DPH §30a, zákon o DPH §29 odst. 1
02
When you are not registered
Eetuj prints you a "doklad o přijaté platbě" — a payment receipt. It is deliberately not called a tax document: claiming you are registered when you are not would be untrue.
03
How the document reaches your customer
Either from your own mail app and your own address — in which case we know nothing about it. Or through us, when it goes from doklad@eetuj.cz and we can show you it arrived. The first is more private, the second is better evidence. Your choice.