Something very similar to what the person on the other side of the table just said:

We hypothesized that in online, virtual formats, negotiators receive better outcomes when mimicking their counterpart’s language; furthermore, we predicted that this strategy would be more effective when occurring early in the negotiation rather than at the end, and should also be effective across both independent and interdependent cultures. Results from two experiments supported these hypotheses.

Mimicry is the sincerest form of flattery.

Through both body language and verbal communication.

Do it early.

In a board or other meeting, you can judge people's attitudes whether they are leaning in or out during the meeting as people show agreement or mental distance. 

Yeah, that's true for pretty much any meeting.

If their heads are in their mobile devices, it's the equivalent of leaning out.

Sincere mimicry seems good; expert mimicry seems ... and feels ... manipulative?  Just spitballing here. 

I agree that the key is to be sincere. Inauthentic copying just makes you look like a faker.

“The secret of success is sincerity. Once you can fake that you’ve got it made.” ~ Jean Giraudoux

The thing about that quote is that I can't tell if it's sarcastic -- or worse, if it's genuine.

I think it's supposed to be cynical humor

I definitely wouldn't take it as advice but I am sure some people do

Heh. Well remember that Oscar Wilde said not to be too sincere:

"A little sincerity is a dangerous thing, and a great deal of it is absolutely fatal."