Update May 3rd, 2015:
My previous answer was for a much outdated platform (latitude doesn't even exist anymore as far as I can tell). Here are some steps for current Google Maps.
Open the Maps application.
Navigate to Settings.
There should be an option to Edit home or work address.
Previous answer below for historic purposes...
I finally figured out how ...
There's an app called ALTitude that does allow you to configure those settings per installation (= per device in your case).
EDIT: Purpose of the app - updating your Latitude status/location base on more fine grained settings (update interval, location source).
ALTitude on Android Market
I'd recommend to configure Latitude to "manual location" ...
According to the Google Product Forums with Google Maps for Android 6.10 the Latitude widget is no longer available. You can view your friends' locations quickly by touching the Latitude icon, or by turning on the Latitude layer in Google Maps.
This is probably overkill and might not even work, but you could root your tablet and install Cyanogenmod, which lets you revoke permissions for individual apps. So you could revoke permission for Latitude to access your location on the tablet, but leave the rest of the app running. This may just cause it to crash, though.
If you go to Settings in your Latitude, under Location Reporting it gives you the option "Do not update your location". Another option is to just turn off the GPS on your tablet, this way it won't give your position in Latitude.
For the current version of Google Maps (7.2.0) set Settings > Google location settings >Location Reporting> Off on your tablet. For more details see the Google help page.
For an older version of Google Maps (6.14.4) I found: Settings > Location settings >Report from this device.
EDIT: the project renamed to OwnTracks
Here's a good, open and decentralized alternative which appeared recently:
A Google Latitude replacement based on MQTT
MQTTitude enables a device to periodically publish its location status to your MQTT broker. If you've previously used Latitude, you can think of this as a a decentralized Latitude.
I had the same problem but didn't see the described GUI on my Galaxy S3. So here's how I did it:
Opened up Google Maps
If on the routes/navigation screen, hit the back button until you return to the default map view with a search bar up top.
The search bar should have 2 icons on the right: one for routes/navigation, and one that looks like a user icon. ...
If you see "Join Latitude" then your latitude options are already "off". If you also now see "location sharing is not turned on", then your account is not set up to share the location information and it will not be sharing the information.
If you don't already, turn off wifi & gps when you are not using them, mpas will be less likely to try and get "...