How are Buddhists supposed to spread Buddhism? Some religions spread through its followers actively seeking out new followers. Some religions do their own thing and inspire others to join. Some religions preach openly to people and let them decide. Some religions cannot be joined unless you know someone who is already in it. Some religions do not have any agenda when it comes to recruiting.

What is the ideal manner in which Buddhism is to be spread to others?
It varies by sect. These questions generally attract a bunch of people who think you are advocating in-your-face Christian style evangelism. 

The patterns I'm going to describe appear to be true for the US, Europe and Russia.

SGI. Works the hardest to gain new members and at least back in the 70s (80s?) was willing to send people out to interact with strangers on the street. It's what makes SGI special, their outreach has reached a broader demographic than just about anyone and what also makes people think SGI is a cult. I'm not SGI, but I think they're far from a cult. They try the hardest to gain members and seem to have had some success.

"Ethnic Buddhist." Some people don't like the word, but it describes communities of recent immigrants who set up Buddhist temples, target recent immigrants, practice in their home language and generally make no effort to involve the "host" community.

State Buddhism. This has been how Buddhism spread in the past. For example in Japan, it used to be that the government assigned you to the local temple. There is some vestigial form of this in modern Japan, most people often don't know what sort of Buddhist temple their family temple is. This means of spread is a nonstarter in the US and Christians are already up in arms about attempts to teach secularized mindfulness in schools.

"(Upper) Middle Class" Buddhism. Generally educated people who are well read and familiar with a variety of world religions and who have chosen Buddhism. This spreads via books and meditation classes.

As for what normatively should be done to spread Buddhism, it's hard to say. On the internet, generally people get bogged down in "respect for autonomy", they have a hypersensitive trigger for seeing someone being forced to do this, or required to do that against their will, or in the case of religion, how teaching religion is some sort of child abuse.

The historical Buddha sent his sangha in every direction to enthusiastically spread the word. Mahayana Buddhism makes teaching (i.e. spreading Buddhist ideas), essentially a requirement, it is mentioned many times in the BNS precepts. Historically Buddhist governments saw no problem with making it a state religion and funding it. I think it would be hard to say Buddhism as a organization, sangha or movement doesn't care about spread, it clearly does. But how to do it, isn't so clear.

Personally, I think Buddhism should be marketed like any other idea and using means that work for a given community in a given time. Some people will buy your book &amp; read it. Some people will go to a book club. Some people read posts on the web. Some people will go to a meditation class if it's advertised in the local community events page.