In Scandinavia most of the iZettle device holders are private individuals and not merchants. How is it possible for a private person to take payments via cards? Is it OK to use a regular payment terminal connected to the land based phone network and charge friends and relatives for monies they owe?

"How is it possible for a private person to take payments via cards?"

In general, if you want to accept credit cards, you need to contract with a credit card acquirer to obtain a merchant account.  The acquirer will check your credit, ensure you have a legitimate business and set a limit as to how much credit card processing you can do.  Credit card processing is very much like applying for a loan/borrowing money - the acquirer settles the proceeds of transactions into your account and then waits to get paid from the institution that issued the card to the cardholder.

With Square, iZettle and the numerous other companies offering acquiring services using a dongle, you don't need to contract separately with an acquirer to obtain a merchant account.  In essence, the supplier of the dongle is aggregating all of its customers under its own merchant account.  Prior to Square, aggregating merchants under one account was very much frowned upon by the Card Associations. Now, Square and iZettle etc. are managing the risk(s) that the acquirer was previously - they are on the hook for fraudulent transactions, chargebacks, etc.  

Although it seems like a fabulous idea to sign up for credit card processing as a micro merchant or individual, there are generally monthly fees and minimum charges.  Most users of these new services simply don't do enough business to justify accepting credit cards.  

Hang on a minute, what about EMV/PCI compliance?  It is OK to swipe a credit card and sign for it but entering the PIN on the screen of the iPhone (or iPad) is not EMV/PCI compliant.

Eventually, the PIN entry must be done on a compliant keypad (ie, the same device that you stick your credit card into).  As soon as the compliance becomes legislated, this will be an outmoded and potentially risky transaction for both consumer and merchant.

Phil - iZettle is an EMV compliant card reader.  That's their value proposition.  And remember, not everyone has the same Chip and PIN rules as Australia.  In the USA, they have announced Chip without PIN.