These tweets stand out a mile because they are not addressed to anyone and typically say something like "That's perfect, can't wait to see you... x" or "OMG OMG! She is such a b@#tch!" - you get the picture.
So how does this happen? Other than potentially being an issue with Twitter (which I am not aware of) its more likely to be user error, particularly when using SMS to tweet.
So.... here's a reminder that when you send an SMS to the Twitter shortcode, it will appear in your public timeline unless you use the correct commands. It's very easy to mess this up when you receive an SMS to notify you of a Twitter DM and you just SMS back to reply! Oopsie.
How to send a DM using SMS
D username + message
Direct Message: sends a person a private message that goes to their device, and saves in their web archive.
Example: d Support Hey support! Thanks for your help
Also if you are using SMS to send Twitter DMs, and your message is longer than 140 characters, Twitter will split it into two messages, but the second part will be in the public stream as it is not prefixed with D username!
Other command examples for using Twitter via SMS are here. they're useful to know if you are ever somewhere with poor cellphone reception and can't get your app to work.
A few other tips:
You can only send a DM to some who is following you. On #newtwitter this is easily signified by the appearance of the Message button on a users profile.
There is a Twitter limit of 250 DMs per day, and #newtwitter only shows the 50 most recent DMs.
If someone deletes a DM they have sent you, it will be removed from your DMs on Twitter. But if you have SMS alerting set up for DMs then you will still receive the SMS when the DM is sent. Clever huh?
Lastly, if you see someone's rogue DM pop up in your stream and they follow you, send them a quick message to let them know (especially if it is potentially embarrassing or damaging for them). Thanks to @rgoodchild for this tip!