It's well-known fact that obituaries are among the cornerstones of genealogical and biographical research, but did you know that law enforcement also keeps track of obituaries? This is because there are criminals who try to assume the identities of recently deceased people, and the police have to prevent them from doing so. To be fair, that;s off on a bit of a tangent, but the point is that obituaries have a wider range of uses than what is usually thought! In addition to use by several fields of professionals, anyone wishing to make a family tree may have to perform a thorough obituary search at some point as well. Keeping that in mind, let's take a look at your options in Maine specifically.
First, gather as much information as possible about the person whose obituary you will be searching for. The absolute minimum is the last name, but you can probably imagine how long it would take to sort through all obituaries of people sharing the same last name (even names you would think are uncommon will likely have hundreds to thousands of matches). So, if the person is your ancestor, try to find someone who knows their first name as well. Also, the place where this person lived is a valuable piece of information, as is the place where they died, if they're different. The reason is that the most likely place for an obituary to be found would be either a newspaper circulating in the place of residence of the person or the place where they died.
Another vital bit of information is the exact date of death. There is a very helpful online resource to help you in that: the US Social Security Death Index. The database spans the period from 1935 to the present day, so if the obituary you are looking for was published no earlier than that year, there is a good chance that you will find all the necessary information in the index. It will give you the full name of the decedent, their birth and death dates, their last place of residence, last social security benefit, social security number, and the state in which the SS card was issued. This is quite a lot of information that could be used in biographical research on its own. This is particularly true when an exhaustive search reveals no obituary that you can use as an information source.
Not all deaths are reported in obituaries. What's more, sometimes they are reported as part of a news story, such as a report on an epidemic or a car crash. These reports can also be used as confirmation of someone's death but bear in mind that you will have to go through a lot of newspaper issues at the library unless you know the exact date of such a report.
Once the essential details about the object of the search have been established, you can go on to see if their obituary is in one of the big digital databases that you see ads for online. Obits for Life is another database, which differs from the rest in that it only lists obituaries supplied by funeral homes, not taken from newspapers, so if you discovered that there was no obituary for the person you are researching, you could try this database. Also, at Ancestry.com you can search death records for the state of Maine spanning three centuries.
Generally, these databases will display the name of the newspaper where an obituary was published, and the date of publication. Some will show you the full text of the obituary itself, which pretty much means the end of your search, unless you want a photocopy of the newspaper page. However, one thing to note about these databases is that they tend to only feature more recent obituaries. If the person you are researching lived and died in the 19th century or the beginning of the 20th, the chances of finding their obituary or death notice in one of these websites are pretty slim.
Your next step, therefore, would be to enlist the help of your local library or the library in the place where your ancestor lived. Libraries store extensive collections of local and national newspapers, even those that went out of circulation a long time ago. This is one of their biggest advantages to electronic databases, and the other one is their expert staff, which can help you in your search.
The Bangor Public Library, for example, has newspaper collections dating back to the 1830s. Although they are not the complete run of the paper, these collections are one more possible source of the information you need. Besides, public libraries also have other genealogical resources, such as cemetery records and death indexes. At Bangor Public Library you can just request an obituary and the library staff will find it for you if there is one. The service is free of charge, except for a reimbursement for any copies that need to be made. Processing time typically takes five to ten working days.
Even if there is no newspaper obituary, the librarian processing your request will be able to verify the event of death with the Maine State Archives Death Index and the SSDI. In that case, if you want a copy of the person's death certificate, you will need to contact the State Archives or the Department of Human Services, depending on the year. For the period 1892 to 1922, the place to go is the Archives, and for death records from 1923 onwards you need the DHS. You can access both the State Archives and the DHS at Maine.gov.
Death records become public only 25 years after issuance, so for a death certificate issued less than 25 years ago you will need to provide proof of 'direct and tangible interest', most commonly a direct family relation. For records older than 25 years you only need to supply a valid identification document. In exchange for a fee, you will get either a plain or a certified copy of the record. If you are looking for an even older record, your best bet is contacting the local authorities in the county where your ancestor died.