Is picture location info retained when I share a photo? When I send a picture to a friend (e.g. via Messages, Mail, or Tweet), is the location information in that photo preserved? In other words, can they see where I took that photo?

What about when I use 3rd party apps such as Facebook?
Yes.
This is a feature usually only offered by professional photo software. Removing metadata (like EXIF or GPS) requires an additional step by the one that wants to share his/her photograph.
Aperture for example offers to export picture as either an original or a version. When you choose to export a version the  meta- and gps-data is removed from the original.
I know from experience that sharing via the following options does not remove these and from pictures:

Mail for OS X
Tweetdeck for OS X
Uploading via a web browser

Here's what the metadata of an export as original look like:

Here's what the metadata of an export as version by Aperture look like. This is an exported version of the same original. The data will differ depending on the export options for a version.