Is it appropriate for reviewer to offer cooperation on a reviewed paper? I am currently reviewing a paper for a journal (area of ecology, organism-environment interactions at various spatiotemporal scales, so including macroecology and evolutionary ecology). Now I am doing a second review of this paper (after resubmission). They submitted a valuable manuscript, but they ran into challenges which are quite difficult to solve. In particular, their statistical population models didn't converge properly; the models would need at least 10x, preferably 20x or 50x as much iterations to run, but they are very slow - taking 2 weeks to run already. I already investigated a bit and found an approach which would make the models run 1000x faster - this would solve the issue. I don't think this approach has been used yet (currently investigating this).
I am about to give them very detailed instructions on how to implement this approach in their models. I am not an experienced reviewer (this is my first paper to review), but (1) I think this is a bit beyond the usual work done by the reviewers? It is also possible that this work will be very challenging for them, since not every biologist is skilled with models and it's much easier for me since I am a specialist in development of new population models. So, (2) is it appropriate for me to offer them cooperation on their paper, with the possibility of becoming a co-author? (3) Should I ask for co-authorship for this? Or, is it better to just contribute and wait if they offer co-authorship to me? If this happens then the editor would probably need to find another reviewer instead of me for the next revision, because of conflicts of interest.
(The review is currently double blinded, so I don't know their names, but I signed the first review and offered my email already, so they know mine.)
Since this is your first review, you probably ought to keep the following in mind:
It is not the job of the referee to fix the paper, it is the job of the referee to provide grounds for the editor to decide whether to accept or reject it. Presumably you have research of your own, and while it is no doubt very kind of you to offer very detailed instructions for how to carry out a new analysis, this is ultimately not what you are tasked to do.
What you can do in your review (which is to be addressed to the editor), is to point out that their current analysis has a flaw, and whether or not you think the paper is publishable with this flaw. If not, then you can suggest that the authors (and not you!) revise the paper using a better method. Here you can give a comment stating your suggestion, and point them to a reference.
And to answer your question concretely: No, it is not appropriate. It would place you in a double role, where the authors might feel pressured into accepting you, to have their paper accepted. If you are very keen on publishing your method under your own name (provided that it is, in fact, novel), you have to wait until their paper is published, and then write a paper of your own. Note that you cannot use privileged information obtained as a referee.
Finally, don't sign your reviews. I would hope that the editor removed your signature. If you don't like giving blind reviews, you should abstain from accepting them, and provide reviews on open platforms which allows this. Like it or not, blind reviews is the name of the game.