If you want Google to report on your XML Sitemap and notify you of any errors and status of Sitemap pages, you must submit it to Google Search Console (formerly Google Webmaster Tools). One of the points mentioned in the Search Console help document under "Why my Sitemap is not listed" clarifies the situation:
Only Sitemaps submitted with this report are listed. Sitemaps sent with the help of google.com/ping or robots.txt are not included in the report, although Google can find and use them.
However, you do not necessarily need to send your sitemap to GSC for Google to retrieve and use it. You could include a Sitemap directive in your robots.txt file. For example:
Sitemap: http://example.com/sitemap.xml
This will also inform other search engines, not just Google. However, it may not be as immediate as submitting your sitemap in GSC because it depends on your robots.txt file being crawled. And, as mentioned, unless you actually submit to GSC, it will not be recognized by GSC and you will not benefit from Google sitemap reports.
Search engines will probably not accept the sitemap if it simply downloads it to the root of your document. XML sitemap files can be called just about anything, so without them being told, search engines do not necessarily know what to look for. If you called him sitemap.xml then you could expect search engines to retrieve it, however, I do not see any request for sitemap.xml in my access logs (for sites that do not contain an XML sitemap), this strongly suggests that they do not.
When your sitemap changes, you will have to resubmit (or notify Google) that your sitemap has been modified. This can be done automatically by whip Google (a GET request), without having to resubmit the sitemap manually. For more information, check out Google's help page on sitemap submission.
