Understand your concern, but it is fairly standard to have to allow others to have a copy of your passport or driving licence in certain situations. When you stay in a hotel you are usually asked to leave your passport with the receptionist for a while, so that they can copy it and pass the copy to the authorities. In Turkey it is a requirement by law (to do with the terrorism act) to register your details at the Jandarmarie which includes giving them a copy of your passport. All visitors to our apartment have to give a copy to our complex manager when they arrive, so she can do this. Perhaps the owner of the accommodation in the Algarve will be doing this in advance in your case. Also when hiring a car the company would take a copy of your driving licence.

Thanks Alfie

We have in the past left our passports with hotel receptions on checking in. In this case the apartment owner wanted us to email him the ID and it was this i was concerned about. He has now agreed we can show ID on arrival so pleased to say we have gone ahead and booked the apartment. Thanks again.

I have rented our apartment to several people over the past few years. Also, I have rented apartments from other people.

I have never heard of giving/receiving passport or other personal information.

The message is clear from the owner your are dealing with - He/She doesn't trust you.

Regarding what the last poster said - if you follow the word of the law you are obliged to take passport details from guests in Turkey.
Don't know about any other countries.
We have had legal advice and have set up our committee jointly with other owners and run our complex according to legal requirements. We were advised that if owners don't comply with the ruling about passports the guests can expect a visit from the Jandarmes! I do know that many private complexes don't have any formal way of running, but would expect that eventually the authorities will start to crack down on this. In Turkey holiday home ownership is a relatively new thing, but gradually legalities are being sorted out. Last year tax inspectors were calling at all the apartments around our village to check if owners were renting out without paying any tax.
Don't think trusting your guests comes into it.

We usually book a private apartment in Majorca and the owner requested a photocopy of our passports - I think it's probably for tax purposes - they will have to keep records of who stayed and when and passports would verify this??????

I could be entirely wrong but I don't think it's anything to do with whether the owner trusts/doesn't trust you!