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