I've read that the online image editor app http://www.picnik.com closes.
How should a competitor react on such a message? What marketing measures are the right ones to reach users that are looking for an alternative?
That is clearly a hot space if Picnik was purchased by Google and Instagram by Facebook. You could setup Google alerts for phrases where people are posting or looking for alternatives. Also do Twitter real time discussion searches for the term, and try to connect individual with prospects and get them to try your software. Buy Google PPC adwords around picnik and see if you can find prospects looking for alteratives. Build an importer from picnik into your app and get people talking about it.
How should you react? Be happy there is one less competitor, be sad Google and Facebook bought the competitor instead of you ;) - Continue to preserver and become a threat to them and make money or get your own buy out.
A similar thing happened to me recently, my biggest competitor decided to shut down their service. I was able to get a big boost in sales for the few months around their closure by targeting their customers. The step I took in the order I think had the most impact were: