A common question we field in our workshops is, "How often do I need to check Asana?" It's a little tricky to handle since different types of roles have different requirements about notifications (i.e. administrative assistant vs. maintenance worker). But our best response is, "How often do you check your email? Probably that much." That's because Asana should become the de facto place to see new tasks and work communication.
But there is often still confusion around when to use Asana rather than some of the other tools you may be using, like email, Slack, phone calls, meetings, etc. What if you need an immediate answer? Does that change anything?
We suggest you create an Asana conventions (or policies) document to set expectations for your team. And part of that document should outline communications protocol for your team.