I've been thinking about this post for ages and was finally inspired to sit down and write it when someone asked me for advice on the matter. So, thank you for the question!
On demand, supplemented nursing sessions with a newborn or young baby seemed obvious enough. When Jacob was hungry, we fed him. When he wanted more, we gave him more. I almost always used supplement during our nursing sessions.
When we introduced solid foods, we assumed that food would take the place of some of the donated breast milk we were continually struggling to find. To our surprise and dismay, it didn't seem to work that way. Jacob took a long time to really get good at eating solids, and even once he did, he never seemed to want any less milk. I sometimes tried to nurse him without using supplement, but he strongly preferred the fast flow that he was used to. Some people suggested not using supplement at night, yet I found that if I didn't he would get increasingly frustrated until he was wide awake. If I used the supplement, I could get him back down to sleep much more easily in the middle of the night (not that it has ever been easy, per se...).
I casually asked friends what they thought I should do, and several pointed out that if a parent with normal milk making capacity nurses frequently, he or she will continue to produce plenty of milk even as the child gets older. In contrast, a La Leche League Leader explained that as the baby gets older, the parent's milk supply naturally decreases. Others noted that babies who were bottlefed typically still get, even as toddlers, a large bottle of milk before going to bed and another one for nap time.
Around the time I was trying to figure out how to proceed, I saw a post in my parenting group from a mom of a 14 month old baby. She was newly pregnant, and her milk supply had disappeared. Her 14 month old still badly wanted to nurse, but this was painful for the mom. They went through a challenging weaning process and the mom started giving bottles of donated breast milk. Her child needed both the milk and the nursing relationship, but it was not possible for her to give. They did the best they could under the circumstances.
I know of another parent who, like myself, is breastfeeding after having had chest surgery. She makes enough milk to have the occasional let-down, but does not have a full supply. Her toddler is two and a half years old and nurses a lot AND gets a significant amount of milk in bottles. Early on, they used a supplementer, but they eventually got to a point where the child no longer wanted it. Their nursing relationship is still very strong.
In another example, the parent of a friend of mine used a supplementer due to her diagnosis of insufficient glandular tissue (IGT), and continued nursing until her child was three. As a toddler, the child would ask for "big milk" when she wanted to nurse using the supplementer, and "little milk" when she wanted to bare nurse. Given the variety of stories I'd heard, I decided to just keep doing what we were doing, since it seemed to work okay for both of us. We switched to putting cow's milk in the supplementer when we could no longer get donated human milk, with no apparent ill effects.
The issue came up again last fall when Diana West came to Winnipeg for a conference. She is the author of Defining Your Own Success: Breastfeeding After Reduction Surgery, and, of course, she is an expert on at-breast supplementation. She stayed with us for a few nights, and was shocked to see just how much supplement we prepared to take to bed with us in the evenings. I felt so busted! If I remember correctly, the much-admired breastfeeding authority said that Jacob doesn't need much milk at this point – it is the nursing relationship that he wants and needs. She reminded me that he was getting some milk from me. She also enthused about how much easier my life would become if I no longer needed to carry supplement around on outings or take it with me to bed at night.
I tried harder this time to wean ourselves from our crutch, but with the same result as before. If I didn't use the supplement, Jacob would quickly get frustrated. The universal and incredibly irritating toddler habit of nipple twiddling got infinitely worse (not surprisingly, this is something babies do to try to get a let-down!), and I again couldn't get him back to sleep at night. To add to the troubles, nursing without much coming out was quite uncomfortable for me, too. Jacob will be two in April, and this is where things stand: we still use lots of supplement at night, during naps, and to re-connect at other points during the day. Sometimes we go on outings without it and I nurse him as needed, but often I take it along for back-up, and then don't end up using it.
As Jacob's vocabulary increases, he is beginning to express how he would like to nurse. He says "nay-nay" for nursing, but also "milk" when he wants to use the supplementer. Sometimes he pleads with me "up!" and "fridge!", until I get up and grab the supplement from the fridge. Other times I ask him if he wants me to get the extra milk and he shakes his head "no" while he is latched on. Ultimately, Diana and everyone else said that we have to do what works for us, and I couldn't agree more. We'll let you know as we go.