Talkativeness relates to several characters.
1. Iago never stops talking. I think we can classify him as a smooth talker too. He regularly talks to the audience about his plans and he certainly talks to Roderigo, Cassio, and Othello to shape what they are thinking and to cause them to act upon those thoughts.
2. Emilia talks in a manner to gossip just as all women do. She is a great representative of women for that matter, but when she realized what had to be said, it was too late:
I will not charm my tongue; I am bound to speak.
My mistress here lies murdered in her bed.
She said this when it was time to tell the truth to Othello about Cassio and Desdemona, the truth being that there was no affair.
3. Bianca and Cassio speak with lovers language, yet their relationship is shallow:
What make you from home?
How is it with you, my most fair Bianca?(180)
I'faith, sweet love, I was coming to your house.
BIANCA:And I was going to your lodging, Cassio
What, keep a week away? seven days and nights?
Eight score eight hours? and lovers' absent hours,
More tedious than the dial eight score times?(185)
O weary reckoning!
4. Desdemona tries to explain things to Othello both before she dies and when Cassio wants to speak with Othello, but is too afraid. Unfortunately, Iago has already reached Othello's ear and has given perceptions that would cause Othello not to believe his wife.
This tale certainly demonstrates that whoever can talk first and with the most persuasive power will be believed. The truth was insignificant until it was revealed in the end and everyone was dying anyway.