Almost never. Files normally stream directly from your cloud provider to the visitor's browser, so the data doesn't pass through your web server. There are a few specific cases where your server is involved.
How streaming usually works
When a visitor opens, downloads, or streams a file, the plugin asks the cloud provider for a temporary direct link, and the visitor's browser fetches the file from there. Your hosting delivers the WordPress page and the plugin's UI, but not the file itself.
The two exceptions
When your server bandwidth is used
1. ZIP downloads of multiple files
When a visitor downloads several files as a single ZIP, the plugin usually fetches each file from the cloud, packages them on your server, and sends the ZIP to the visitor. During that operation the files pass through your server, so they count against your hosting bandwidth.
For Box and Dropbox, the API can sometimes serve a multi-file download directly without server-side packaging, particularly when no file filtering is active in the module. In those cases your server is not involved.
2. Gallery thumbnails (Box and Dropbox)
For Box and Dropbox, the plugin downloads the source image once to your server to generate a thumbnail. The thumbnail is cached and reused for future visitors, so the cost is one-time per image. The originals are not stored permanently on your server.
For Google Drive and OneDrive/SharePoint the thumbnails are delivered by the cloud API directly, so this step doesn't apply.
3. Google Drive fallback streaming
In some cases the Google Drive API doesn't return a direct download link. When that happens the plugin streams the file through your server as a proxy. This is a fallback: the plugin uses direct links wherever the API allows.
What about my cloud provider's bandwidth?
Yes, file transfers do count against your cloud account's outbound-traffic budget if your plan has one. See Does the plugin have usage limits? for how that works per provider.