Hi,
In theory you don't need to syncronize the redirects (and you might not want to) because Umbraco will pick up redirect as an when the content is published.
e.g if you rename a page on stage, the local stage redirect will be created by umbraco but when you publish it to live then Umbraco will once again create the redirect for you.
this can be a good thing, because if you rename a page multiple times on stage, and then only publish it once to live the live site will have the single redirect that is the change and not all the intermidiate steps (that where never live).
e.g
change a page from page -> page local to -> page published stage will look like this