There is a setting in Mathematica that controls whether it can access the internet. Go to Preferences -> Internet Connectivity and uncheck "Allow the Wolfram System to access the Internet". Disabling this will disable some features that depend on internet access, such as Wolfram|Alpha queries.
This setting can also be controlled by the $AllowInternet global variable.

When doing Wolfram|Alpha queries, Mathematica can send additional information to Wolfram|Alpha about the current session to aid the interpretation of natural language input.
This can be controlled in Preferences -> Internet Connectivity -> Wolfram Alpha Settings. The default setting is "Ask before sending". With this setting, Mathematica will present a dialog box like the following before sending the information:

Opening "Details" will show the information that is being sent with the current request.
When searching the Documentation Center, in the top right corner will be a small piece of text saying "$n$ results on all Wolfram sites", unless you have disallowed Internet access. This number $n$ is produced by a WRI server that receives your search, as well as your $LicenseID and your computer's $MachineID. The full query URL is:
http://search.wolfram.com/lucene/numberofhits.jsp?query=<your query>&collection=tryonall&mathid=<$MachineID>&license=<$LicenseID>
It does not seem to matter if you substitute incorrect or nonsensical values for the mathid and license fields, or omit them completely--you will get your result anyway. Presumably this means that WRI does not really care about the contents of these fields, so perhaps their reason to include them was as a tool for defending against possible DOS attacks on their server.
The obvious implication of this is that one should not search for sensitive search terms with Internet access enabled. However, the likelihood of revealing any confidential information through searches in the Mathematica documentation seems realistically quite remote, unless you have a tendency to make rather unconventional searches.
The DemonstrationsTools` package, which supports the authoring of demonstrations, contains vestigial code for the DemonstrationTemplateOpen function (which opens the authoring template notebook) that obtains the template from the Wolfram server, if it does not exist on the local machine, using a query that includes the $LicenseID and $MachineID as part of the URL.
This code is vestigial because, directly below this code, the function is immediately redefined to avoid the use of Import, and the redefined version does not include these fields any more.
The paclet manager, which is responsible for documentation updates, providing the data sources for the *Data functions, and updates to some packages (e.g. CUDALink`), sends the following to the paclet server when it communicates with it:
$SystemID
$LicenseID
$MachineID
- the language of this copy of Mathematica (English, Chinese, or Japanese)
$ActivationKey (i.e. the extended $LicenseID)
I have not checked to see whether or not it matters that these values are present and correct.
Information sent
In most contacts with Wolfram|Alpha or the paclet servers for the *Data family some data about the user and his platform is transmitted. This includes:
- App id
- MMA release id (10.0.0)
- System ID (Windows-x86-64 etc.)
- License number
- Machine ID
All in all not very frightening if you have a legit installation, but it allows WRI to track you all over the world whenever you use Mathematica (but many web sites can do that too).
tcpdumpshowed nothing. I would very much like to see who and why claimed that stuff got sent even then (I very strongly suspect it's BS, but one should be able to find out) – acl Jul 15 '14 at 14:39tcpdumpwithout root. I am sure it's possible, but would very much like to see who and why claims it. Because if they are sending stuff, they are putting a lot of effort into looking completely incompetent (by sending stuff in plaintext etc) on the surface. Or they're doing it to avoid suspicion, – acl Jul 15 '14 at 14:42WolframAlphawants additional information it will prompt you. If they were going to do it they would disclaim it somewhere. – mfvonh Jul 15 '14 at 14:48tcpdumpthen perhaps you should stop using mma. python/scipy is also just fine, is free, no license limitations, and open source so you can inspect the source to your heart's content. – acl Jul 15 '14 at 14:52