No. The 1860 census is not copyrighted. And the digitized image should be attributed to where you found it; a good citation for Ancestry or Family Search would be:
1870 U.S. Census, County, State, population schedule, Post Office, page # (note whether it's penned or stamped), dwelling #, family #, line #, Name of person; digital image, Ancestry.com (http://www.ancestry.com : accessed 3 October 2019); citing NARA microfilm publication #, roll #.
If you find the image at the National Archives - the NARA has its own preferred citation format, and you should follow that.