My 2 cents on the op's original question, based on what I've experienced, yes it does make a difference. I am not a working professional (in this area), but have a handful of friends who are/want to be. And from my interactions and conversations with them, and their job hunts, knowing Avid, and FCP has been essential. I've never used Lightworks, so I can't/won't comment on it. But what I've heard from my friends, when they were looking to get hired, the businesses, in their hiring add/requirements, listed specific software they were looking to hire someone to use. Friends who had experience with an NLE, but not the one the job was for were given little to no consideration.
If you're looking to work as a freelancer, I'm sure more of APE's post are even more valid. If your post workflow can't match up, then it can hinder your ability to land a job.
I have a good deal of experience on FCP (if I have to knock something out fast, that's my go to NLE, because I know if best). I'm switching over to a more Adobe workflow (because I'm getting more in 3D animation, and compositing, and the back and forth between Premier and AE is easier). My future plan is to gain a working knowledge of Avid, because I would like to transition from my current field to a new one. And I think having a working knowledge might help me get my foot in the door in some fashion.