How do Classified Sites make money?

1. AdSense

Worth mentioning about Classifieds is that they can be smoothly presented on empty searches, so as not to display empty searches - instead of just having it has related content.

2. Display

Display has traditionally been an important revenue source for most digital media, but some classifieds offer display in more creative forms. High traffic volumes and users searching on city and category levels make it possible to sell targeted display on geo and interest. For certain brands who have reached a winner takes all position (e.g. Blocket and http://Finn.no) display has proved to have a higher growth on mobile where profesional stores/boutiques pay for being part of search lists in banner formats, sometimes with a slight delay for increased exposure.

3. Listings

There are numerous variants of monetizing listings

By number
Blocket is one of the few sites that charge for every single listing in all categories. Most sites, do it only in a few (Craigslist) or like eBay Kleinanzagenonly charge for heavy usage i.e. more than 5 ads. 

By pro/private
Another dimension instead of by number is charging differently for businesses, which is quite common among the mature sites (http://Subito.it,http://Mobile.de, http://Avito.ru etc.). A fair, common and straightforward division, but which often results in creating a behaviour where sites have to moderate to find “hidden professionals” trying to cheat the charges.

Premium listings
The most common form of way to monetize classifieds used by almost all of the big players Naspers, eBay, Schibsted, Avito, Kinnevik/Rocket and others are:
Bumps or enhanced or premium. Listing are bumped to top of the search results. Either fixed as a sticky posts or just by renewing the timestamp.
Design highlighting with border, bold text or the like.
Links or placements of the ad in other areas than the search result.
Time extension or prolonged listing time.

4. Auxiliary Services / Affiliates

A pretty standard setup is to work with related affiliate link on listing pages which may even enhance the user experience and not come across as bought media. Horizontal classifieds mainly keep to personal finance links (loans, insurance etc.), escrow or shipping.  On markets where listings are free or easily accessible (Verticals and the US real estate market in particular) these are more prevalent.

5. Data

Few marketplaces openly talk about selling data, but especially for strong verticals meta data of prices and trends is quite valuable for banks, insurance companies and other players.

6. Premium Rate Numbers

Offering private numbers, unique to the listing, to sellers is fairly common on many markets, but is mainly offered as a quality service but some give take a small cut, like Blocket.

8. Selling Leads to Businesses

This business model somewhat diverges from the general idea about classifieds, so this is a stretch, but still deserves a mention. http://www.myhammer.de, http://www.uShip.com and many more charge per number of leads delivered or fixed matching fee