I was born in Ireland, so through no particular skill on my part, I speak English as my first language. Generally speaking, we Irish don't carry too much baggage when we travel. We've never got round to conquering any other nations (at least not directly), so, if they've heard of us, generally other countries tend to be quite well disposed toward us. This makes the Irish NEST's life a good bit easier. Our accent can sometimes pose a challenge for learners (I had to live in the US for three months before I could pronounce my THs) but once you take the edge of it a bit, slow down and avoid words like rashers, nobody really has a problem understanding us. To make things even handier for us, you get nonsense like this coming up every so often.
So all in all, I would argue, that on the spectrum, your Irish NEST has things a bit softer than most (apart from the cowboyism still a part of the industry here).
Being in this lucky position, you tend to wander along thinking to yourself what a wonderful teacher you are - how great your lessons are, how engaged your students were, how wonderfully you explained the difference between the present perfect and the past simple. And maybe this is true - maybe you are a wonderful teacher. But by reading the posts I have mentioned above, and, more fundamentally, by sharing my life with an incredibly talented and inspirational nNEST, I have come to realise that good teachers are good teachers, irrespective of what their first language is.
There are really good arguments here as to why it makes no sense to favour a NEST over an nNEST; the Wikipedia page on the topic has some good links; and this webpage is a nice reminder that its not just NESTS teaching English out there.
For my own contribution to the debate, I thought I might be practical, and consider what teachers can do. I believe that the people who run schools and hire teachers are acting from a practical concern - that students will favour the school with the most NESTs. I honestly don't know if students actually believe this, but if they do, then where does that leave us. Do we accept that? Does one school use this to manufacture a competitive advantage over the school who hires the best teachers, irrespective of mother tongue? I think these are questions to be considered.
As teachers, I would (preachily) suggest that there are a few little, practical things we can do to support our colleagues.
1. Don't distinguish between NESTs and nNESTs. We're all teachers.
2. If you hear of a job up for grabs, recommend the person you think most suitable, regardless of where they come from.
3. Praise other teachers in front of students. Again, regardless of where they come from, if you think a teacher is good, don't be afraid to say it in front of students (they'll still love you).
4. Think critically about the idea "students prefer native speakers". Do they? Who says? How wedded are people to this notion?
5. Be kind to other teachers. Chances are they're having a tough day, native or not.