There is no difference in how ISE operates, when you involve public cloud. You can put all or only some nodes in public cloud, and have some on-prem. Since ISE nodes communicate via L3 IP, your only concern is ensuring that all the nodes can talk IP to one another.
Why people put ISE in the public cloud is not clear to me. There is no feature benefit. The only reason I can see is that these customers have no choice, because they no longer have any on-prem locations to host a physical ISE server, or host an ISE VM. That's a strategic/commercial decision made by any customer. The cost and operational benefits have to be compared.
I don't see there being too much operational benefit, since, unlike elastic services (like web servers) we don't spin up ISE servers every couple of days. ISE servers get built once, stay running and unless there is a need to rebuild them, we only patch them. And patching ISE in public cloud is no different to patching on-prem ISE.
