Don't say hello in chat
Also see NoHello.com. Async communication is in many ways the reverse of sync communication, and this is true for niceties as well. While it's important to be cordial and friendly, you can often invert normal social conventions and put information up front and chit-chat afterwards instead of the other way round. When a conversation takes place over hours or days, then:
Hi John, could you share that widget we discussed last week with me (hi, by the way - how are things) ... wait Hi Mary, here it is (going great - how's there? Looking forward to the weekend tomorrow)
is much more efficient and respectful of each other's time than
Hi John, ... wait Hi Mary, how are you? ... wait Good thanks. How's there? ... wait
Default to including too many people than too few
To avoid three way conversations, always include anyone who might be relevant from the start. People in async environments are bottlenecked by responding to emails, slack threads, etc not by reading them so no need to 'save people's inboxes'.
Make it clear when you are owning a decision or expecting someone else to to own it
'Shared' responsibility is inefficient and often people have strong opinions over small things. Be prepared to own certain decisions, deliverables, or conversations and don't spend excess effort getting to consensus (bias for action).