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.