English Translation
They shook hands with each other.
Although there are no examples of this in the corpus, I think you can also say “Si sono stretti la mano.”
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ReversoContext has Si sono stretti la mano in a couple of examples. In CM search for la mano I found some even distribution of the sentences containing dare and stringere. Yet, CM did not offer the exact si sono stretti, but it does not make it wrong I guess.
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Interestingly, when I stayed with a family a Firenze, and was introduced to Alessandra e Francesco, I just nodded and said “Hi” but Francesco laughed and reached forward, “Ma dai, signorina, in Italy we always give the hand”. Always remembered it along with “Dammi del tu”.
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Yes, also the kiss 
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Magari non mi deve baciare sentence is a recent addition to the kiss collections in CM given the current state of events. Purtroppo.
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Sadly even in the pre-mask “good old days” Brits were not comfortable at close-quarter greetings. As a “tactile”, the hand and the kiss is fine with me.
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A firm handshake is quite enough, thank you 
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Not too firm, mind, and not for too long;-DD
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English Translation
They shook hands with each other.
I have seen “stringere” mostly used when it’s from one person’s POV (he shook my hand). This seems to have mutuality. Even in English you can say either “we shook hands” or “I shook his hand” and they have different shades of meaning.
I am reading “Il Disprezzo” by Moravia right now (excellent!), and there are many examples of rather unwilling hand shaking - in that case I don’t think “dare”: would be used.