Scirra's multiplayer feature is quite diverse, and most likely capable of what your looking for ... but without some experience with construct itself, I can say it will be quite the challenge.
Scirras multiplayer can run serverless, such as in it does not require a real server with web server and or database capabilities.
It requires a signalling server, but that is provided by scirra.
The Scirra multiplayer "server" or better said host element, is mainly the controlling factor determining who is actually doing what.
Using positional synched objects, the objects from the host are mirrored to the peers connected.
If you are looking to store various data long term, I would suggest using ajaxing and communicating with a php/mysql backend.