Ooops, I just read "starter" and inmediately thought about the electric. Why would the kickstarter give any trouble at all?
As dgfischer pointed out, the kickstarter has a spring that presses a serrated tooth wheel against the gear wheel. When the kickstarter is at rest, another indentation in the oil pan fully disengage the gear wheel to the kicstarter wheel. When you kick it, the spring presses against the gear wheel but if the engine start to run, the orientation of the tooth makes them to spin and not catch the kickstarter wheel, I don't know if I have explained myself well. So you either have worn wheels, or spring not fully strong.
Now the bad news: you will have to take the engine out, turn it upside down and remove the lower cranckase to service the kickstart.
