etiquette - Should I notify people I mention in the acknowledgements section?

I recently finished a paper in which I received valuable feedback from two professors (neither supervisors nor coauthors) at some specific points, which ultimately leaded to localized significant improvements. I highly appreciated it and of course I added explicit mentions in the acknowledgements section stating their suggestions.

My question now is: I hardly will see again these professors for a long time. I thought about sending an email to them commenting on their inclusion on the acknowledgements... But then I thought that it should be something understood as obvious, without too much interest for them or even spammy. So, shall I communicate their inclusion in the acknowledgements section? If so, what will be the correct etiquette for such an email?

EDIT: As @StephanKolassa comments, another important and highly related issue is what is the best timing for communication, if this is considered adequate:

• First submission


• Acceptance


• Publication (early access or traditional)

Answer

shall I communicate their inclusion in the acknowledgements section?

Yes, why not? It helps at building connections and improving relationships. Besides, if the acknowledged people are from a different field than yours, they might never know you've acknowledged them otherwise.

what will be the correct etiquette for such an email?

I don't think there is any need to be overly formal. Typically I would write something simple, along the lines of (names and facts have been changed to protect the innocent):

Dear John,

thank you very much for helping me with the problem of packing holes. I've included an acknowledgment in a paper I recently wrote on the topic.

The paper is titled "Packing the unpackable and stacking the unstackable" and has been submitted to the Transactions on Painstaking Stacking. You can find a preprint of the paper at the address (link to, e.g., arXiv).

Kind regards/Sincerely/Cheers,

Massimo

The above example is meant to be sent just after the first submission. In more critical cases, when I'm not sure whether the acknowledgment would be well-received or not, or if I think I might have written a wrong detail (e.g., the affiliation), I typically send a copy of the paper before the submission, asking for feedback.