Predecessor Task 1 and successor Task 2. You want assignees to Task 2 to receive Start notification on the day when Task 1 finishes (and Task 2 starts through a dependency).  

Proceed as follows:
​

1. Set a Finish-to-Start dependency between Task 2 and Task 1.
2. Set the lag to '-1' for the dependency to ensure that the start date of T2 will be today if T1 finishes today.
3. Make sure you have Start type notification enabled in the notification rule for the respective type of activity (task, issue or request).
4. Make sure T2 has either no start date or one different from today.
5. Close T1: This will change T2's start date to today's and trigger Start notification.

For this to work, Task 2's start date must change, from no date or from a different date, to today's.