How to approach a researcher regarding his latest publication? There is a group that I've been interested in for a while because we share some of the research interests and because of the very ingenious approach used by its PI. Lately, they published a paper in exactly the same field I work in (but with different models objects).

I feel really eager to react, and I have some considerations about their results, parallels with our work, ideas for future research etc. However, I'm not actively applying for a post-doc, nor do I have any specific (that is, supported by funding) propositions for collaboration. However, I would like to get the conversation started, ideally leading to some kind of common project. It's not a typical cold-call, since


the paper in question cites my group's work, 
I've met the PI at a conference and had a brief talk, and
I studied alongside two of his postdocs at a practical course.


We are separated by the ocean, so the communication will have to be via e-mails of some such.

So, how should I approach? E-mail, commenting on ResearchGate, writing to those postdocs first (even though they weren't involved in this particular peace of research)? Specifically, how can I formulate the concluding, "so what" part of the letter?
I have some considerations about their results, parallels with our work, ideas for future research etc.


So use this as a starter. Write/ask about their paper and get a discussion going. In my experience this will very often automatically lead to a collaboration of some kind, like a follow up paper or a whole joint project.

As an alternative I would recommend that you talk to your supervisor, what he thinks about the group and if he would think starting a collaboration would be worth a try. Maybe he will then start to initialise the communication.