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Privacy of text messages

Don't give them access to your phone, and use a screen lock.

There are also apps you can install which can password-protect access to other apps (e.g. your message app), and I think some message apps might let you define particular threads as "private" (though the one I use does not, so this is anecdotal).

If by "outside sources" you mean the spy agencies, forget it - if they want access they'll get it through your carrier, not through your phone.
 
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