This kind of functionality is definitely something for which you should use the Flag module (at least). Using a flag that can be set by individual users (not a 'global' flag). Together with a few modules as further detailed below.
Message Stack
Have a look at the Message Stack, which consists of these modules (quotes are from the module's project page):
Message:
The Message module is the core of the message stack. It enables logging and displaying system events in a number of different use cases. Events that are recorded over time are sometimes call activity streams. Exportable messages subtypes can be created for different use cases with custom fields and display (view) modes.
Message Notify.
This module provides a method for sending a message via a notifier plugin. Message Notify comes with plugins for email and SMS and may be extended to other transport mechanisms as required.
Message Subscribe.
With this module, users who subscribe to content will be notified when events occur that involve that content. The module leverages the Flag module, which provides a subscribe functionality to users.
These are the basic components and dependencies of the message stack:
If you want to have a closer look at the Message Stack in your own environment, then experiment a bit with the Answers module (disclosure: I'm a co-maintainer), together with the Answers Notification sub-module, which allows logged in users to subscribe to selected questions, so that they receive notifications (using the Message Stack) when Questions receive Answers.
Rules and Views
The tutorial about "How to allow users to manage their own Message Stack messages?" explains how to use the Rules and Views modules as a natural complement of it. And how, optionally, you can also use the Menu Badges module to show the number of Unread Messages via a (red) counter (similar to like on iOS).
Video tutorials
If you're not familiar (enough/yet) with Rules, checkout the 32 (!!!) great (+ free) video tutorials Learn the Rules framework, organized in 7 chapters.
There is a similar set of 8 video tutorials about the Flag module, often considered as a natural complement to the Rules module.
And a set of 30 video tutorials about the Views module.
