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As a peer reviewer, I sometimes feel there are issues in the manuscript but I'm not sure how they should be corrected/addressed, partly due to my inadequate expertise in those exact issues.

I also sometimes find some words/terms are incorrectly used, but as a non-native speaker I can't easily suggest alternatives.

Should I just point to such issues anyway, or ignore them since I couldn't suggest solutions?

Does "peer review" mean you have to re-write the material, or just point out where the flaws may be and the author is meant to sort them?

I suggest it is the latter, so point them out and expect the author to edit / correct or justify what they meant.

Yes, you should point them out. You should point out solutions to problems when you know what the solution is but, at the end of the day, it's the authors' responsibility to write their paper, not yours.

A story (and many other intersting ones) I once heard from an editorial board member of Physical Review Letters, who gave an overview talk on the editorial process of that journal at a conference, was that sometimes it happens that the reviewer switches sides and becomes a collaborator of the authors they reviewed initially.

While at first this sound strange (certainly did to me at that time as a young PhD student), I think this is more appropriate then outlining new solutions (which is nice but not necessary) the authors did not think of. I also don't think 1-2 major revisions are a good spot to discuss/recommend in-depth new solutions to a manuscript. It's common to request further data analysis/evaluation or additional measurements. But if there are major flaws in the manuscript/methodology, you should point to it, but personally I would advise/vote then to reject the manuscript for this reason.

Concerning grammar and language mistakes: The associate editor can reject publication of a manuscript, even if the peer reviewer don't vote for further revisions (due to content or language level). It's not your duty to improve the language and associate editors regulary recommend commercial english editing services, when the language level is below the standards of the journal. Personally, if I see a lot of spelling and grammar mistakes, I don't point the authors to every single one, I mention 2 or 3 to the editor and the rest is his job, not mine, especially if the manuscript is multi-authored.