I can only speak for executive speechwriting (not political), but I broke in by creating relationships with people who source speechwriters.
Some execs have full-time speechwriters who write for them all the time. But a majority just get freelancer writers for big events. Someone on their staff (or their agency) finds and hires these people. If you are known by those people, and have a good rep, you start getting jobs.
It's actually hard for pro journalists to break into exec speechwriting, because the writing is not complex (in fact, the POINT is to dumb it down). The real value you provide is getting into the exec's brain and teasing out the story, coaching them on positioning and points, etc.. So it's more like psychology, understanding both the client mindset and the audience mindset, mixed with a business background.
