>Why am I having the perception that, from Power BI Desktop, when we publish to Power BI service, and we click on Open File.pbix in Power BI, that this goes to a Microsoft server? Isn't there a way to keep our data local?

There're two options; use Power BI pro and it gets your data from your data source 8 times in a day, use Power BI enterprise, install a Power BI server and your Power BI reports will run and pull your data from that server. I used both of them, second is more expensive.