When you set up a Pulse SMS account, it uploads all of your contacts to the web (end to end encrypted). When you are composing a new conversation from the web app or desktop app, Pulse SMS will search through those contacts. If some contacts are missing or not available to message from the web, there is a topic on this page around missing-contacts that discusses this.
When composing a conversation from your tablet, it is pulling contacts from the Contacts app on your tablet. The tablet does not read from the contacts stored in Pulse SMS' online database.
As a side note, conversation titles do not have anything to do with the contacts that are stored in the cloud. Titles are assigned to conversations when that conversation is created, from the phone app. They are not looked up dynamically.
