First, if you have some problem, and you migrate everything from one mac to another, the problem usually comes along.
Second, access to "local network" is not necessary for most any applications. It certainly is not needed for browsers. Most people have an empty list of applications there.
Create a new user, and test. Does it show the same problem? If it does, then this is probably due to some party modification (system extension, agent, daemon). If not, then it is something specific within your usual account.
Please do tell us which is the case.
In order for us to try and figure out what is causing you trouble, we need to know more. Please run Etrecheck and post its full report here.
Use the "additional text" button and paste the report into the text box.