Route servers do propagate prefixes from one peer to all their other peers. At least, that's how route servers typically operate today.
In figure 1, the networks do not peer directly, but all maintain BGP
sessions with AS 25, the route server. Note that each AS has allprefixes, and the AS path is a hop longer because AS 25 appears in it.
Figure 1: BGP sessions towards a route server: http://d39z3tvn9akifj.cloudfront.net/ccie/wp-content/uploads/2015/02/bgp-sessions-1024x499.png
Under normal circumstances, the traffic would now flow through the route server, making running a route server on a big internet exchange a non-starter. However, BGP is smart enough to recognize that all the route server peers are connected to the same subnet (the internet exchange peering LAN). So unlike it does under other circumstances, BGP doesn't update the next hop address. This means that when AS 456 gets the route to 184.108.40.206/24 from the route server, the next hop address isn't the route server's address, but the AS 123 router's address. As such, packets don't flow through the route server, but rather, are directly delivered to the right AS, as shown in figure 2.
Figure 2: Traffic flow when peering with a route server: http://d39z3tvn9akifj.cloudfront.net/ccie/wp-content/uploads/2015/02/route-server-1024x502.png
However, there are also some networks that have an open peering policy and will peer directly with anyone, but don't peer with route servers, because that way they don't control their peerings. For instance, when there is an issue with a peer, it's useful to be able to temporarily shut down the BGP session with that peer. If the peering happens through a route server, you'll either have to wait for the route server operator to take action or shut down the BGP session towards the route server and impact lots of other peers, too.
Most networks prefer to peer through the route servers exclusively when possible, while other networks like to add direct peering in addition to the route server peering. The argument in favor of a route server only policy is that it keeps the amount of work and the number of BGP sessions to a minimum. On the other hand, adding direct peering has the advantage that if there's an issue with the route server peering, there's still the direct peering.
The BGP session towards a route server is configured exactly the same as any other BGP session used for peering. The route server itself can also be a standard BGP router, or a Unix (-like) system running BGP software. The route server configured to allow prefixes from each peer to propagate to other peers. A regular router performing route server duties will include its own AS in the AS path for prefixes it propagates, as shown in figures 2 and 3. This makes paths learned through the route server a hop longer, so paths learned through direct peering will be preferred. However, it's not uncommon for route servers to be set up to leave out their own AS number in the AS path propagated to peers, so there's no impact to AS path length.
Source: Peering with Route Servers - BGP Case Study