When you set up a Pulse SMS+ account, it uploads all of your contacts (from the Contacts app of your primary phone) to the cloud (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 on the web/desktop app, please check this article to try to resolve the issue.
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 names/titles do not have anything to do with the contacts that are stored in the cloud. They are assigned to conversations when that conversation is created from the phone app. They are not looked up dynamically.