I'm not talking about when they speak their first words, or even their first sentences, but the age where they can carry on a decent conversation with (say) kids of the same age. What age does this happen?
Well, it's a gradual process.
What are the milestones for speech and language development?
The first signs of communication occur when an infant learns that a cry will bring food, comfort, and companionship. Newborns also begin to recognize important sounds in their environment, such as the voice of their mother or primary caretaker. As they grow, babies begin to sort out the speech sounds that compose the words of their language. By 6 months of age, most babies recognize the basic sounds of their native language.
Children vary in their development of speech and language skills. However, they follow a natural progression or timetable for mastering the skills of language. ...
Here's some of the milestones which relate directly to speaking:
1 to 2 Years
Uses some one- or two-word questions (“Where kitty?” or “Go bye-bye?”)
Puts two words together (“More cookie”)
Uses many different consonant sounds at the beginning of words
2 to 3 Years
Has a word for almost everything
Uses two- or three-word phrases to talk about and ask for things
Uses k, g, f, t, d, and n sounds
Speaks in a way that is understood by family members and friends
Names objects to ask for them or to direct attention to them
3 to 4 Years
Talks about activities at daycare, preschool, or friends’ homes
Uses sentences with four or more words
Speaks easily without having to repeat syllables or words
4 to 5 Years
Uses sentences that give many details
Tells stories that stay on topic
Communicates easily with other children and adults
Says most sounds correctly except for a few (l, s, r, v, z, ch, sh, and th)
Uses rhyming words
Names some letters and numbers
Uses adult grammar
Selected items from Speech and Language Developmental Milestones, NICD (USA), Updated September 2010, based upon How Does Your Child Hear and Talk?, courtesy of the American Speech–Language–Hearing Association.
So essentially the answer to your question - when can they carry on a decent-ish conversation - is "somewhere between age 3 and age 5".