This is indeed hard to find information, but I'm going to give you a quote from Daphne Fautin about the reproduction of anemones. But before I do I'll point out that there really is very little information that I can find about the specific reproductive habits of the normal hosting anemones that we all know and love. Also there seems to be a bit of debate about whether or not an anemone can bring in food from the water column as in filter feeding to some degree. I'll continue to do some research on both reproduction and filter feeding in anemones up to and including getting a hold of Dr Fautin (yeah I've got her email addy)
by Fautin:
All coelenterates reproduce sexually. An individual of some species may produce both eggs and sperm; host anemones appear to have separate sexes, with an individual being either male or female its entire life. The typical coelenterate pattern is that of most marine animals, one that is fraught with dangers and uncertainty -- release of eggs and sperm into the sea, where fertilisation occurs and a larva (a tiny animal looking nothing like its parent that drifts in the sea) develops for several days or weeks before settling in an appropriate habitat. Many species spawn in response to an environmental cue such as a full moon or low tide so that eggs and sperm are in the same place at the same time. Typically, marine animals produce millions of tiny larvae, but the world is not overrun with them, proving that very few survive -- usually just enough to maintain a stable population. The rest of the larvae serve as food for a sea full of potential predators. Finally, the surviving larvae must find an appropriate habitat (how anemonefishes might do this is discussed in chapter 4).
We do not know if host actinians follow this pattern. There is a bit of evidence that in at least some species, the eggs are not released, but are fertilised inside the mother (this is not especially rare in corals and anemones; sperm enter the mother with water that is constantly being pumped in and out, and which carries food and oxygen also), where they grow to be released as tiny sea anemones. What is certain is that we seldom see small individuals of most host actinians in nature. However, it is not unusual to find large ones with ripe eggs and sperm. Therefore, we believe that successful recruitment must be rare. Very few eggs may be fertilised, or few larvae may survive, or larval settlement may be difficult, or young anemones may have high mortality (perhaps especially when they are too small to harbour fish). The apparent rarity of successful reproduction is also biologically consistent with long life.
In addition to sexual reproduction, some coelenterates undergo asexual reproduction. Entacmaea quadricolor is one of these. A polyp can divide longitudinally, resulting in two, somewhat smaller individuals, probably within the space of a few days. Each then grows to an appropriate size, divides, and so on. All descendants of the original anemone (the result of sexual reproduction) form a clone, a group of genetically identical individuals. In this species, each polyp is relatively small, but clonemates remain next to one another so their tentacles are confluent, and the associated anemonefish apparently regard them as a single large anemone.
This is so mainly for shallow-water individuals; those in deeper water grow large, and do not divide (see chapter 1). Several other species of actinians also have two different reproductive modes: small animals that clone and large ones that do not. This appears true of Heteractis magnifica, too. In the center of its range (i.e. in eastern Indonesia, on the Great Barrier Reef, in New Guinea), it occurs as single, large individuals. To the east and west (i.e. in western Indonesia and Malaysia, and in Tahiti), several to very many small individuals of identical colouration are typically clustered together, appearing to be a single large (or huge!) anemone. Based on their shared colour and their proximity, we infer that they are clonemates.
I can agree that there simply isn't (that I know of ) enough documentation about specific anemone reproduction. I do know from my own experience that the E.Quad sure does divide and produce clones, I've got about 2 dozen. But rarely do I address a post that states that my LTA is dividing, and when I do it wasn't splitting.