OK. Lets see if I can not lose you here... If you Tweet at AmericanAir:

1. Anyone who visits your Twitter profile will be able to see the Tweet. For example, if you look at the FlyerTalk Twitter... https://twitter.com/FlyerTalk You can scroll down the profile page and see all of the people that they have Tweeted at and all of the other Tweets that they have posted.

2. Any of the people following you WILL NOT be able to see your Tweet at AmericanAir IF you start the Tweet with "@AmericanAir". For example, this Tweet won't show up in the feed that all of FlterTalk's 14k followers see... https://twitter.com/FlyerTalk/status/367346012320309248 Twitter treats Tweets that start with an ampersand as being public conversations between two people (or in this case 6 total accounts).

3. Any of the people following you WILL see your Tweet at AmericanAir IF you start your Tweet with anything other than an ampersand. For example, this Tweet showed up in the feeds of all of FlyerTalk's 14k followers... https://twitter.com/FlyerTalk/status/368115575194542080

4. Anyone who visits the AmericanAir profile will NOT be able to see your Tweet at AmericanAir if they do not respond to your Tweet publicly.

5. Anyone who visits the AmericanAir profile WILL be able to see your Tweet a AmericanAir if they do respond to your Tweet publicly. https://twitter.com/AmericanAir/with_replies

6. Anyone who searches Twitter for any of the keywords that you include in your Tweet to AmericanAir will be able to see your Tweet. For example, if you wanted to tell AmericanAir that they sucked, these people did it before you could... https://twitter.com/search?q=americanair%20sucks

*No one can access DMs except for the 2 accounts exchanging them and then of course anyone who has access to the log-ins for those accounts (ie: bigger airline/hotel accounts have multiple reps handling the Tweeting)

*You can always delete the Tweets that you send, but unless the other party also deletes their Tweets as well, half of the exchange will still remain on Twitter.

TL;DR: If you're hoping to avoid airing dirty laundry in public on Twitter, your best best to to send a short Tweet starting with an amersand+theaccountname at the account that you want to contact letting them know that you have an issue that you'd like to discuss and asking them to follow you back so that you can have a DM exchange.