Private people who want to share selectively with a tight-knit group of friends and family should probably just stay away from Twitter. Twitter is a great social network for public figures (and people who want to be public figures), because it essentially functions as a broadcasting platform. But if you're looking to make or keep relationships, it's not the most suitable network for your needs.
The good news is that protecting your privacy on Twitter is relatively easy--mainly because it's a fairly simple social network. The bad news is that you have to choose to protect your privacy on Twitter from the start: You can't retroactively protect previously posted public tweets. Furthermore, protecting your privacy on Twitter can make for a somewhat boring Twitter experience.
To protect your tweets on Twitter, go to Settings > Account > Tweet Privacy, and check the box next to Protect my Tweets. Once you've activated this setting, all of your future tweets will be visible only to your approved followers. This restriction applies to @tweets and replies, too: You can reply to people who don't follow you, but they won't be able to see your tweet unless you approve them as a follower. Consequently you can't use Twitter to communicate with people outside your approved follower list unless you unprotect your tweets.
There's no way to limit your past tweets (public tweets always remain public)--and if you unprotect your account at some point, all of the previously protected tweets will become public, and will stay public forever.
Ultraprivate social networks
If you feel that Facebook and Twitter are too public, you may want to take a look at private social networks. The following social networks are designed for close-knit groups who really want to connect with each other--not social butterflies who want to broadcast their lives across the Internet.
Couple: Formerly known as Pair, Couple is the ultimate private social network--a smartphone-based network designed expressly for couples. In fact, you can only have one friend on Couple: your significant other. Couple features a timeline that's a bit like a souped-up text message exchange--you and your partner can add photos, reminders, important dates, drawings, and videos, along with regular text messages.
Family Wall: If you're looking for a slightly larger social network, FamilyWall helps you keep track of your entire family. At this private, Facebook-like social network for families, you can add dates and events, photos, videos, contacts, messages, and even Foursquare-style check-ins. You can also add "Family landmarks" such as schools, doctors, and fitness centers.
23snaps: Instead of posting photos of your children on Facebook or Instagram, try posting them to 23snaps, a smartphone-based social network that lets you create a unique, private online photostream. 23snaps lets you add photos, videos, and status updates to a special photostream of your child (you can add a stream for each child) and then share those photos with your friends and family. Another option is to co-manage a 23snaps account with your partner, so you can both add photos of your kids.
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