The first step to finding print or microfilmed copies of Maine newspapers offline is Chronicling America from the Library of Congress. The database offers information on almost 1,500 newspapers published in Maine, the dates and places of publication, and the locations of bound or microfilmed copies.
Copies of the Lewiston Daily Sun (1890-1989) and Lewiston Evening Journal (1860-1979) and Lewiston Sun-Journal (2005-2006) may now be viewed online in the Google News Archive.
Copies of the Bangor Daily News (1970-2007) are also now available in the Google News Archive.
Other Maine newspapers have been digitized, and are available to view behind subscription paywalls.
Ancestry.com's Historical Newspaper Collection has searchable scanned copies of:
NewspaperArchive.com has copies of:
The NewsBank "Early American Newspapers" series offers digitized copies of 21 Maine newspapers from the 19th century, and may be accessed with membership in the New England Historical and Genealogical Society, among other organizations.