she happened to mention that she had been "looking out for the death announcement" in the local paper, but hadn't seen one and wondered if my friend knew that this was something that people are supposed to do
Not meaning to be rude to your DM, but she has no right whatsoever to tell bereaved people what they are 'supposed to do' in dealing with the busy aftermath of the death of a loved one.
Death notices in the paper are a 'traditional' thing and are still quite popular for people who want to put them in, but there's absolutely no shame or faux pas if you don't for any reason. As a PP said, they often cost upwards of £100 or more, at a time when all the funeral costs also have to be found, added to which people may have to take unpaid leave to deal with all the practicalities (and not to mention to grieve).
They may also wish to grieve privately (the deceased may have been a very private person too) and putting a notice in the paper might lead to receiving a lot of unwanted and intrusive calls from people you vaguely knew a long time ago passing on their condolences, wanting more information (and then possibly using it as a link to unhelpful, upsetting personal anecdotes) or, sadly, even just wanting to grief-surf and maybe make it all about themselves and THEIR perceived loss, as some people sometimes do.
There may be some people on the fringes who otherwise wouldn't know, but everybody close would already have been told or found out on the grape vine. As you say yourself, she was 'looking out for' the death notice, so she already knew about the actual death. Some people (especially older people) do like to look through the death columns to see if old friends/acquaintances have died, but they have no absolute right to demand that there be a notice for everybody who's died, just so they can look through and say "Oh, I see Lily Peters has died" to their spouse before immediately moving on to reading the next notice/page. Some almost see death-spotting as a hobby and gloss over the fact that their old acquaintance in whom they have a passing interest and whose death gives them a bit of gossip fodder is a grieving, often devastated family's beloved mum, dad, brother, sister, GP etc.
To summarise, putting a death notice in the paper is a nice thing to do IF you want to; it is nobody else's business whatsoever if you DON'T do so, for whatever reason.