So if understand correctly the local leader perk will allow me to access the storage from my main workshop to all other cleared areas that have modification tables ( armor, weapons, railroad) so I can modify anything anywhere?

I believe you still have to build supply lines before it connects them. Local Leader just lets you get access to building those.

yes, you have to "assign" a settler to go from your current settlement to another. I believe the option is below on the UI after clicking on the settler. The only shared content will pretty much be the junk or anything stored in your workbenches.

Local Leader Rank 1 allows you to assign supply lines. Once you assign a supply line to link two settlements, the junk, food, and water will be shared between settlements (i.e., accessible for people's needs as well as your crafting).

The workshop storage itself is not actually shared. Only the resources are shared as needed (i.e., people having a needs shortage in one settlement can get it from another or you choosing to craft items, buildings, etc.).

Stuff like weapons and armor are NOT shared, so that is important to keep in mind.