I’m having the same problem (for about a week). I did think it was due to having the app open at the same time, but that’s proven to not be the case (had an online transaction this morning - over Wifi - which definitely didn’t trigger notifications - and delayed personal emails meant I wasn’t at all certain the merchant actually took the payment!) [added: a Google Pay payment also failed to flag up a Monzo notification - this time via 3G]
Checked Monzo App->Settings->Notifications->Transactions and payments ARE enabled.
Checked Android Settings->Notifications->Monzo and ‘Block all’, 'Show silently and ‘Override do not disturb’ are all off and ‘On the lock screen - show all notification content’.
Current App 2.0.0 beta branch - last updated 13 hours ago (and I think there was a 3rd version earlier in the week which didn’t have notifications). Android 7.1.1 on Nexus 6.
[Perhaps we could have an ‘Advanced’ setting under Account with things such as ‘Test notifications: See what a notification would show on your phone’ which may help diagnostics]