The Local Council has never had a rigourously defined role, but their purpose, in general, is to represent TSP's gameside community. The initial plan was for Councillors to make decisions on things such as spam, polls, telegrams, etc., while also working with the MoRA to promote gameside activity. It's worth bearing in mind that this was during a time before Regional Officers - essentially the Delegate was the only person who had any control gameside, so the LC could have no real power.
Since then, the game has changed (with Regional Officers now being a thing, the LC actually has power to make polls/suppress posts etc. by themselves), but the spirit of the LC hasn't. It still exists as a way to help include the gameside members of TSP in the running of the region, and in regional activity.
In the past the political aspect of the LC has predominantly been communicating between forumside and gameside: promoting gameside interests on the forums, and letting the gameside know what was happening on the forums. At one point, the LC had a block vote, worth 1/6 of the total vote, that it could cast in any forumside legislative vote as a representation of the gameside. However, the current system is for the LC to decide its own laws independently of the forums, while not having any say in the forumside legislation which doesn't affect gameside.
The full list of LC regional powers is as follows:
Appearence - can change the WFE, the flag, the pinned dispatches, and the region's tags
Communications - can send region-wide telegrams, change the welcome telegram, and change who can send recruitment telegrams for TSP
Polls - can make gameside polls
It's worth noting that while the LC has these powers, some of them (such as changing the welcome TG) cannot legally be used without permission from the Delegate.