Does one have to practice engine failures in flight training?
From a general aviation standpoint here in the US, yes. Engine out landings and general procedures are practiced during your PPL training. No particular failure is trained for because there are really only 2 situations, the engine is running or the engine is not running. The procedures for different planes vary but generally you set for best glide, pick a touchdown point (airstrip if you can) and head for it. Once in route you can attempt a restart if you can identify the problem and solve it. on some planes you can feather the propellor and such. 

That being said that generally applies to single engine planes, in a twin engine plane there are a few more situations depending on the design of the plane. In a twin you may have a "critical engine" as such engine outs are practiced with that in mind and the procedures are adjusted appropriately.

As for the why... 

There are lots of things that can take out an engine like a bird strike, a mechanical failure, a fire etc. The fact of the matter is that it does happen  all be it not that often, you would be a fool not to prepare. 

Anecdote: 

When I started my training I got curious about piston engines cutting out so I began to ask around. Youtube videos aside, I was only able to actually find one guy who knew a guy that once lost an engine. Engine manufacturing has come a long way since the advent of airplanes and engines are simply more reliable than they once were. Here is a great article that touches on some of the twin engine points about reliability. While it does happen and you should never assume "this wont happen to me" engines are much better than they were 50 years ago when some of the requirements were made for the tests and training standards.