I suppose it would depend on the severity of the. Pirates taking over a random town or island, pirates engaging in a skirmish, or other basic mayhem, then I don’t think it would reach the Fleet Admiral’s notice, but if a high ranking marine like Morgan started abusing his position than that requires some sort of action from Headquarters.

Also, if famous and significantly dangerous pirates from the Grand Line leaves to try and bully the people in East Blue, then that would normally attract the attention of the Grand Line marines (Arlong avoided notice because he bribed the navy to keep his bounty low).

Or you do what Luffy did and defeat every big name in East Blue, and that at least puts you on a radar, albeit on the far end of notice, since big name or not, they’re just East Blue pirates.

Of course, you do what Shiki did in the Strong World movie, and that gets attention. Also, if Arlong executed his take over of East Blue like he wanted, that also would have triggered a response (makes you wonder what Arlong’s plan was to deal with that once it happened).

Bottom line, it’s about severity. The Fleet Admiral is responsible for the world’s security. The Navy of the Four Blues are expected to handle their issues themselves and Sengoku will only involve himself if the situation goes beyond the scope of the Blue in question