Tough to really answer as each has it's advantages and disadvantages. I value data ownership so I will always lean towards self-hosting, but that isn't always practical as we are talking about a community, not one person or even a small group of highly technical users.
Maintaining reliable access to your user's email will be a constant source of frustration. The reality is, email is old, and fragile and the last 15+ years companies have spent enormous amounts of energy reducing spam. They have also been consolidating the space of who is a trusted sender. So unless you have a large set of users who are willing to fight spam blockers, a willing moderation team able to handle communication problems between your list and sometimes unresponsive postmasters, you might not want to continue to rely on this.
I maintain a small web forum and I have to constantly monitor my logs and be in contact with email postmasters because I use email verification for the registration process. Even still, emails from my board end up in spam boxes. You could forego email verification of your web forum and use moderator verification instead if your group isn't large or if there is some other way to verify your users. However, now communication has become more active rather than passive, so this might not be something that is compatible with your user base. As they will need to remember to go somewhere to catch the latest messages without some sort of reminder or notification via email.
Going to Google Groups or another hosted email system will massively remove a lot of these administrative headaches. These companies are places your users are probably already spending a lot of time, they also are a trusted sender by most postmasters so you will see good receive rates, but now you are tying your fate to that of company who's interest might not be aligned with yours or who may end up changing the rules or taking away the service you have begun to rely on for communication.
You could go with a hybrid approach, pay for an SMTP service and funnel your email through their system, but I have found that unless you have extremely low volume this is unreliable for free, so you should be prepared to pay for this service and get a dedicated IP so that you have some control over the IP's status on blocklists.
Utilizing other social media should be the lowest on the list of options, LinkedIn, Facebook, etc. This is really not that different than just hosting a forum, however if your user base spends a lot of time on these platforms or relies on them for other uses, then this point would be moot. However, as with the Google Groups option, you are now tying your community to a company that might not have their interests aligned with yours.
Good luck in your search.