Three formats cover nearly every real conversion
PNG when transparency or crisp edges matter - logos, icons, screenshots, diagrams. JPEG when it is a photograph and universal support matters more than anything else. WebP when it is going on a website and you want the same picture at a fraction of the weight. Conversions between those three are the overwhelming majority of what anyone actually does, and every one of them is something your browser can already encode.
The honest boundary is the input side. Your browser decodes what it decodes: an iPhone HEIC, a camera RAW file or a layered PSD may not open here at all, and when that happens the page says which format it found rather than failing vaguely. Those are precisely the cases where a server with a hundred decoders installed is the right answer.
The upload is not just a privacy question, it is the slow part
Converting a five megabyte PNG takes a browser a moment. Uploading it, waiting in a queue, and downloading the result takes considerably longer than that on any normal connection, and repeats in full every time you decide the quality was wrong and want to try again.
Local conversion changes the rhythm of the work. Change the quality, look at the size, change it again - each iteration costs nothing, so you converge on the right setting rather than accepting the first acceptable result because another round trip is not worth it. The same applies to converting a folder of forty images, where forty uploads is the entire cost of the task.
Why the converted file is sometimes bigger
Converting a JPEG to PNG almost always produces a larger file, and people reasonably assume something has gone wrong. Nothing has. PNG is lossless, so it faithfully records every compression artefact the JPEG introduced, and it is designed for flat colour and sharp edges rather than the continuous gradients a photograph is made of.
The other direction has a worse trap. JPEG to PNG does not recover quality - the detail was discarded when the JPEG was made and no later format can bring it back. A JPEG converted to PNG is a bigger file containing exactly the same information, which is why converting to PNG for quality is a step that costs bytes and gains nothing.