TL;DR: This is an argument that (1) Talking is a major way that COVID-19 transmits from person to person (at least when people aren't wearing facemasks and goggles, and possibly to a much lesser extent even if they are), right up there with coughing, and probably more important than touching contaminated surfaces; (2) The public is not generally aware of this, at least where I am; plus (3) Sketchy ideas for what to do about it.

Epistemic status: Low confidence, open to criticism.

Background: Why think that talking (at least without facemasks) is a major transmission risk?

This NIH video shows how talking creates droplets and aerosols, just like coughing does, and strongly advocates that everyone should avoid face-to-face conversations. They also made a follow-up video showing that facemasks help, even the kind of DIY facemask made from old T-shirts.
This CDC webpage emphasizes respiratory droplets and avoiding close proximity, adding that "contaminated surfaces or objects … [are] not thought to be the main way the virus spreads." The Chinese government guide concurs.
This article by a domain expert argues that the main way people catch it is probably if someone is talking or coughing nearby, and agrees with the CDC that touching contaminated surfaces is probably less of a big deal, albeit with considerable uncertainty from the lack of data.
- (She gets really pedantic in arguing that we should use the terminology "short-range aerosols" for what's going on, while the CDC, WHO, and everyone else seems to call the same thing "inhaled droplets". I really don't care about the terminology. Everyone agrees about what's going on. But to be clear, the term "aerosol" usually refers to long-range aerosol transmission, like measles, and there seems to be a consensus that this type of traditional long-range aerosol transmission is not a major part of COVID-19 transmission, for reasons discussed here, including observations about who catches it from whom. (ETA: ok, maybe the word "consensus" is too strong, see here.)
As jimrandomh and others have argued, there is a lot of asymptomatic transmission—so it’s not just people coughing.

Super-spreader events seem to involve singing indoors and/or face-to-face conversations

Two known COVID-19 super-spreader events may have involved the infected person singing indoors for an extended period:

First, the Skagit Valley Chorale case:

...The virus has devastated the Skagit Valley Chorale, based in the rural valley north of Seattle... Of the 60 people who attended a March 10 practice, 45 have developed symptoms and 27 so far have tested positive, officials said. One of the group’s members has died, another has been hospitalized and others have struggled to overcome their illness.

Ruth Backlund, a co-president at the Skagit Valley Chorale, said the group was monitoring public health guidelines at the time of the practice and had asked people to stay home if they showed even minor signs of illness. The group gathered in rows facing a piano and a choir director. They were all in individual chairs and had space to keep separated. Ms. Backlund had made sure there were extra soap dispensers in the bathrooms for people to wash their hands.

"Nobody was sick. Nobody touched anybody. Nobody shook hands. Nobody hugged everybody like you might do in a group. There was none of that," Ms. Backlund said.

Second, maybe the case of the Shincheonji religious sect in South Korea. I don't know the details, but in general, "Attendants remain seated on their knees throughout the service, chanting Amen and singing en mass".

Other super-spreader events seem to be consistent with face-to-face conversations with asymptomatic individuals:

The Connecticut party presumably involved face-to-face conversations, though it seems that few details have been published. The infected person is said to have been asymptomatic.

The Boston Biogen Conference involved people milling around with hors d’oeuvres and drinks—again, presumably consistent with spread by face-to-face conversation. It seems that nobody was coughing there, though I could be wrong.

What can we do?

I think public health messaging has not gotten through about this. I propose a motto / meme:

When someone talks to you, they're spitting on your face!

This message would help people understand viscerally why they're being asked to stay 2 meters apart. It would also help people understand viscerally why people are telling them to make and wear DIY facemasks & goggles. (I just made myself a mask yesterday!)

Beyond that, awareness by itself would be really valuable in directing people's decisions and actions. For example, where I live (USA), many of my local businesses (grocery stores, take-out restaurants, etc.) are going above and beyond in frequently sanitizing surfaces, non-contact e-payments, staff washing hands and wearing clean gloves, etc. etc. ... yet shopping there still involves a face-to-face conversation with the salesperson (who does not have a mask)!!

I want every boss of every essential business to tell their employees: Don't try to start a face-to-face conversation with your coworkers or customers (especially without facemasks), it's just as rude and reckless as coughing on them, or pushing them into a busy street. I want a cultural norm that says that if someone tries to start a face-to-face conversation, the other person is entitled to run away!

Again, I could be wrong altogether ... and if this is a message worth promoting, I'm not sure how to get lots of celebrities etc. to say "When someone talks to you, they're spitting on your face!", instead of "wash your hands" (which is still a good idea but everyone has already heard that a million times).