No, it’s not an endless loop waiting to happen, the plan here is to use Logstash to parse Elasticsearch logs and send them to another Elasticsearch cluster or to a log analytics service like Logsene (which conveniently exposes the Elasticsearch API, so you can use it without having to run and manage your own Elasticsearch cluster).
If you’re looking for some ELK stack intro and you think you’re in the wrong place, try our 5-minute Logstash tutorial. Still, if you have non-trivial amounts of data, you might end up here again. Because you’ll probably need to centralize Elasticsearch logs for the same reasons you centralize other logs.