When you buy a book from Amazon it says on the Detail page if the Lending option
You are allowed through Amazon's Lending program
to lend the book 1 time for 14 days. The person you are loaning the book to has 7 days to pick it up and 14 days to read during which time that book is inaccessible in your archives. This is a one time deal for each book that has the lending option enable.
If you go to a previous purchased books at the top in yellow banner with black text it says: Loan this book to anyone you choose.
What these sites do is hook you up with a person to borrow your book. You then loan them the book via Amazon's site. It's perfectly legal. You can also request from someone for them to loan you a book.
If you want to do it the old fashion way. Amazon Kindle Forum particularly the romance forum has a Loan Me Thread
where you leave your email address and the book you are wanting to be loaned. This is a more messier way to do it. You are opening yourself up to spambots or worms so use a email address you don't care about. A site like Lendle or LendInk would be much safer which is why we need them.