Wow, is this relevant to me!
While my family is only 3 of us, I am quite content using a private (and self-hosted) social network. I ended up using Gnu Social (http://gnu.io/social/).

Here are the reasons for wanting a private network:
* Wanted to teach my young daughter how to behave and what to be prepared for (i.e. cyber bullying) while utilizing social media...BUT...In a safe environment before going out and facebooking with her friends and such. I think of this environment as my daughter's staging social network before she deploys to production. ;-)
* Wanted to talk with my wife via private mechanisms that aren't as easily mined for data. Signal (the app) has actually replaced this need quite nicely: https://whispersystems.org/blog/signal . But its nice to know we can still use our private social network for private conversations. Though admit my stack is absolutely no match for NSA-type intrusions.
* Wanted to keep MY OWN ARCHIVES of a timeline of important family events. If i lose an important photo/note, its on me, and I don't have to rely on Facebook, Twitter, etc.

Challenges I faced (and you might want to consider):
* My daughter is already getting fed mis-information from her (air-headed) young friends that facebook, twitter and the like are the only social networks in existence (or the only ones that matter in their eyes). It took some teaching to inform my young daughter that the web is open, and that Daddy can manage our very own network, thank you very much. ;-)
* My wife has a habit of relying on conventional text messaging - she is by no means a tech guru, so tends to forget that we have our own social network and text messaging isn't needed between us 3.
* All the usual challenges that come with self-hosting stuff: uptime/availability, software updates, maintenance, etc. are your responsibility. Like spider man's uncle Ben said: with great power comes responsibility.

Here's some info which may help you decide what would work best for your scenario (understanding i don;t know what your skills or expected time investments are):

* I tested out Friendica - which was pretty straight-forward to install (just simple PHP stack) but seemed a little cumbersome to use. To be clear, it does alot out of the box, though now thinking, maybe it was too much GUI, widgets, etc. for my small family...And, what I needed was simple service, like a private twitter. I am interested when the red matrix project (same originators of Friendica) becomes more mature. Ultimately, i killed this Friendica installation and moved on.

* I tried pump.io, which actually is pretty cool...but because all my stuff is on a small VPS infrasturcutre, pump.io seems to want to be on its own vps...So, to install it on infrastructure that is shared with my personal site, etc. just seemed to consume too much time. i abandoned this - just didn't have time to play sysadmin. Much like red matrix, here's another project that has a bright future (at least i hope). Just understand: pump.io is a protocol and not a single app/server, so until the killer server setup is created (to implement said protocol), I think adoption - for folks who can but don';t have time to play sysadmin - might be slower than it could be. By the way, from the mobile client perspective, I played around one of the more popular clients: Impeller (http://impeller.e43.eu), which actually worked really well. Before I setup my own test server, i tried it on another person's server (that opened up public registrations) and found impeller to be pretty fast/nimble.

* Finally, I installed and have been using Gnu Social for many, many months now. The install was a simple php stack, some-what straight-forward. The only annoying thing is that the documentation - at last months ago - was slightly inconsistent between website and what gets downloaded with installation bundle. But in actually once that inconsistency was surpassed, the install scripts work pretty well. It was the simplest of the installs compared to the platforms i referenced above. An important point: what actually kept me on this platform so far has been the mobile client (an app called &status: http://andstatus.org). While the platform itself seems pretty solid (I can't speak to scaling issues, considering I have only 3 users!). But this mobile client is pretty freaking fast/nimble; and i don't think its my little VPS! There was another client called mustard which worked ok too. I suppose because the underlying platform is decent enough, but the clients really shine here. In this case, with my wife being more used to native styles of text messaging apps, &status really helped with her adoption. In any case, Gnu social is what i have been running with, and so far so good.

There are many other options (buddycloud, diaspora, etc.) which i did not try, but which might be quite worthwhile depending on your use-cases.

Good luck!