Best method of remote tutoring? My tutor and I are exploring ways of doing lessons over the internet, and to our surprise, there isn't an immediately obvious way to do this on lichess or chess.com. What we were looking for is an analysis board that we can both move pieces on at the same time with maybe a chat room too (though we could just call each other, so that part isn't a deal breaker). 

Is anyone aware of such a site/program? Worst case, we can do a screenshare and I can just watch him move pieces on his screen, but that is definitely less than ideal. 

Surely this problem has been solved since I can't imagine we are the first to encounter it. I hope I am just overlooking something obvious.
Have you tried lichess studies ?
They can be used exactly like that. Let me give you an overview of how it works.
First you need to create a study, which you can do by heading to  https://lichess.org/study/ and clicking in the green plus button.
Multiple Members
As the What are Studies page explains, you can invite people to your study:

Let me quote the page that explains what these studies are:

Invite friends to analyse that new trendy opening. Invite your coach to teach you something. Invite 20 students to your lesson. Why not?
If your study is public, anyone can review it; invited members can chat, and invited contributors can update the study.
If your study is private, only invited members can review it, and invited contributors can update the study

Moves, Symbols and Arrows
The study members can create moves, or delete them. Any move anyone does changes the board that all other members see. Any move can have arrows to point possible moves or threats as well as symbols which are the red circles that normally are used to point key squares:

Comments
Any move can have comments, and these will be identified by who made them:

Annotations
The moves can also have the common chess notation:

Chat
There is also a chat, in case anyone wants just to say anything that isn't a comment to a move:

Chapters
This feature is solely available in studies, and allows you to have multiple chapters over the same study. It is useful in many scenarios like when you want to try several different ideas with quite some depth and extended analysis while keeping the main line clean. You could have the analysis of all your games in a tournament within the same study, and turning each game into a chapter.

Engine
As you would also probably be expecting, there is a feature to use an engine evaluation any time you need it. At the time of this answer the default engine is Stockfish 9.