When does lactation occur?
Stage I lactation occurs from midpregnancy onward; stage II is the onset of substantial milk secretion associated with having given birth.


  In humans, stage I occurs at approximately midpregnancy... the gland is sufficiently differentiated to secrete milk, but secretion is held in check by high circulating plasma concentrations of progesterone and, possibly ...estrogen. 


You ask,


  ...is anything produced then? 


Yes. The mature primed breast produces colostrum, containing sodium, chloride, immunoglobulins, lactoferrin and other substances. 


  I’ve heard things about milk leaking out of breasts during pregnancy...


Not at all common that I know of. I'm sure one can manipulate the breast to get a small amount of milky discharge, but that's a big leap from leaking. More common (again, that I'm aware of) is galactorrhea associated with early oral contraceptive pills, prolactinomas, and other conditions.


  the breasts seem very heavy and laden with milk during parturition.


The breasts mature during pregnancy in preparation to make milk; that's why they get bigger. They can get even bigger when they are actually full of milk. It may not be as obvious in a human, but in milk animals (goats, cows, etc.), well, a full udder is pretty obvious.

Different researchers divide lactogenesis into a different number of stages; this is a simple two-stage approach.
Lactogenesis: The Transition from Pregnancy to Lactation