Selecting a file gives the page access to it
A browser's File API lets a page read files you select with a file picker or drag onto it. It can use that data to display a preview or perform an edit. Sending the data to a server is a separate action that a website can implement. The presence of a file picker alone does not tell you whether a tool uploads files.
For a tool that performs the conversion locally, the image work happens on your device. Read the explanation for the particular tool you are using. A privacy statement about one converter should not be assumed to apply to every feature on a website.
Further reading: MDN: reading and uploading selected files
Separate the image from the website visit
A tool can process an image locally while its website still loads pages, fonts, or other resources over the network. Local processing describes what happens to your selected image; it does not by itself establish that a visit is anonymous or that no other data is collected.
Look for a clear privacy policy and an explanation of file handling. Useful details include whether the original is uploaded, whether an account is required, and what happens to generated files. Treat broad claims such as “completely private” as a reason to look for specifics.
Understand what a network check can show
For a closer look, use your browser's developer tools. Chrome's Network panel records requests while it is open and lets you inspect request details. Open it before selecting a harmless test image, then watch what happens during processing and download.
This can help you investigate the current workflow, but an empty-looking request list is not a complete audit. Filters, timing, and later actions affect what you observe. Test with an ordinary image first, and consider your browser extensions and the device you are using when deciding what to process.
Further reading: Chrome DevTools: inspect network activity
Check the image before it leaves your device
Local conversion does not remove information visible in the picture. Review screenshots for names, addresses, open tabs, and notifications. Check reflections and background details in photos. A smaller file can still reveal exactly the same personal information.
Do not assume a format conversion removes all metadata unless the tool documents and verifies that behavior. Review sensitive files with a suitable metadata viewer. Finally, choose a clear download location and remember that sharing or uploading the result is a new step with its own file-handling rules.