I also have a site on wordpress.com. You know, wordpress.com is blocked in China, so I just used that site as a backup. It works perfectly in my last devastation of data loss. I recovered my lost posts from this backup site.
However, there is a problem. When I post a blog, I have to do it twice. One on nanzt.info and the other on nanzt.wordpress.com. I have to feel lucky I have only two mirror sites. Therefore, I decided to test this plugin, namely Push syndication. The latest version v2.1 is only availale on github.com. (There is a 2.1-wp branch, grab it as a zip file, and upload to your own-hosted website.)
Be aware, this version from the github only works with php 7.1. The reason is the scripts contain a function named mcrypt_*, which has been deprecated in php 7.2+.
We have to create a new wordpress.com application when setting the plugin. The important thing is the redirect url which should be properly set (the url is provided in the settings page of this plugin). Upon success, we can see OAuth information provided by the wordpress.com application page, on which information such as client id, secret, redirected urls, etc are shown). Fill client id and secret to the plugin setting page and then save the changes. After that, click Authorize at the end of this page. It will go to a wordpress.com page to connect your wordpress.com blog. When it succeeds, some important information will exhibit, including Access token, blog id and blog URL. We need those information at next steps.
Then go to the sites menu on the left barside. Create a group first and add a site where the access token, blog id and URL are required to enable this site. It is quite easy.
When we begin to write a post, do not forget to check the associated site group that will be syndicated on the rightside of the page. The same content will simultaneously appear on all the sites. However, the images uploaded will not be transferred to the associated sites.