Check in / out in a web Document management System

We have developed and implemented a document management system using Windows Forms.

The check out/in feature works as following:

1. User A Check out a document
2. The application asks the user where to save the document (checkout location)
3. A copy will be downloaded on the specified location on the user's pc.
4. Now, if the user edits the document, the modifications are saved locally on machine.
5. Other users cannot make modifications to the document, they can only view the version on the server (until User A check in the document)
6. User A, can open the document, the modifications are saved, he can now check in the document, the document is uploaded as new version to the server.
7. Now other users can view the updated version of the document and check it out if needed.

Now, we are in the process of developing the web version.

• Where i should keep the checked out documents?
• If locally on the user stations, how should i save it and upload to the server on check in, taking into consideration that the application will run on a web browser.
• Or in the web solution we have to keep the checked out documents on the server.

I am asking about the design and strategy about how check in / out document in web document management system.

How does it work in SharePoint (does it download offline copies on the user's desktop, how can we do a similar solution)?