Are we going to be able to use dart not only on the client/browser, but also on our server?
Write some code for a server e.g. Apache, accept browser requests, query a database and return a response to browser? Maybe create web services?
Edit: here is the link to a video where server side Dart is confirmed:
http://news.dartlang.org/2012/03/video-josh-bloch-talks-about-dart.html [min 36:00]
Short answer yes.
Longer answer. The Dart VM already contains a HTTP chat server example that shows how to use the dart:io HTTP API to implement a web service. In general the dart:io library contains functionality that is meant to be used in server side programming such as sockets and file/directory handling.
Its still very early stage but already now you can play around with some early attempts of implementing connect/node style applications using the Dart SDK.
As for communicating between the client and the server you will have to use classic REST/HTTP mechanisms, but in the future (when reflection is implemented in Dart) you will be able to serialize actual Dart objects and thus share them between the client and server.