If I use Text Blaze to copy a body of text, is that text copied and stored on Text Blazes server? It doesn't appear to copy to my computers clipboard.
Hi, and welcome to the forum! 
Snippet processing and insertion happens locally on your computer. Text Blaze does not transmit content from commands like {clipboard} in snippets to our servers.
For advanced uses cases, you can use connected snippets (Text Blaze | Connected Snippets Overview) which can transmit data. But you must explictly enable and configure that feature to use it.