Publishing a paper - not enrolled in university I've never delved deeper into this but I'm still curious: do I need to be enrolled in a PhD to publish scientific papers?

I have a MSc but I regularly read research papers of not-so-great-a-quality and think "I might write better stuff than this". It doesn't happen with the majority of papers but the point is: I don't think those authors are all smarter than me.

How does a 'private citizen' not enrolled in any PhD course publish a research to a journal?
No, you do not need to be enrolled in a PhD to publish scientific papers. You don't need to be a professor either. It's common, but by no means mandatory. For example the game Arimaa was invented by Omar Syed. He published a paper introducing the game and has never earned a PhD.

As for how to publish - it's no different from those who have PhDs. Go to your target journal's website; it will have instructions for authors on how to submit the paper.