For this question, I sent mail to three differnet Groves (Red Earth in Atlanta, GA; Silver Birch in Australia; and Ocean's Tide in Rhode Island) whose rituals I have not attended, and two solitary members of ADF (Jeremy in Chicago and Brandon in Japan). I would like to thank each of them for their responses to my questions.
First, we ask why we make these kinds of offerings. A Praise Offering is an offering that may be physical or not (material objects may be offered, or prayers, praises, songs, or other such things may be offered), done in an ADF ritual. Most responded that we offer as part of a *ghos-ti- or reciprocal relationship with the Kindreds. Red Earth is the only respondent that didn't specifically mention reciprocity, but they did mention that the praise offerings "make public our bonds with the Kindreds."
Other reasons given include Red Earth's "a chance to learn of beings and powers we may not individually have encountered before;" Ocean's Tide's "we think they're swell and they deserve it," and "to recreate/continue some old religion tradition;" and Brandon's "it's tradition--in the early ADF days praise offerings were the main offerings, or the only offerings."
In general, a lot of the response seems to revolve around the idea that this is the right thing to do for those with whom we have a special relationship, or with whom we might desire such a relationship.
Each of the Groves mentioned or insinuated offerings that were non-material. While the Solitaries also mentioned this, their responses tended more toward the physical offerings. Jeremy stated that most of his offerings go into Lake Michigan, and so he strives for small and biodegradable offerings. Brandon mentioned: "It doesn't feel the same in a solitary setting. I feel much better giving appropriate material offerings. The exception is on the Skira and the Eleusinian Mysteries, when I read out part or all of the Homeric Hymn to Demeter."
Red Earth does nine rounds of offerings, with the first being to the Deity of the Occasion and the rest being open to any Ancestor, Nature Spirit, or God/dess. Silver Birch simply makes offerings that they believe are appropriate to the receiver. Ocean's Tide has three rounds of praise offerings, one to each of the Kindreds. Their Senior Druid, Anthony, has been requesting that people make an effort to offer praise to the Deities of the Occasion in the "gods" round, but doesn't require it of anyone.
Most of the respondents make their praise offerings during the "Key Offerings" section. The exception is Silver Birch, who indicates that they "make them at the point the receiver is encountered for the first time."
Ocean's Tide makes their offerings right after the key offerings to the Deities of the Occasion, and before any seasonally-specific meditations. Red Earth "set aside a portion of the ritual between the invocations of the Deity(ies) of Occasion and the Main Sacrifice and dedicate that time to Praise Offerings." Both Jeremy and Brandon make theirs during the Key Offering section, with Jeremy adding, "after the gods have been called and the reason for the rite explained, but before asking for anything."
This has a wide spectrum, probably due to confusion over what the Clergy Council tried to do in 2006 (the CC asked people to remember the Deity of the Occasion in praise offering sections, but the announcement that went out was poorly distributed and not well received, pointing to a major disconnect between the CC's mission and the CC's ability to manage that mission). While the original intent of the praise offerings was to make offerings to the Deity of the Occasion only,¹ it has developed in many areas away from that and spread into (in some cases) a "free for all" offerings section. I have seen Christ given offerings in two ADF rituals, as an example.
All but one of the respondants mentioned making offerings to the Deities of the Occasion. Brandon is the only person to mention that the DotO is the only person praise offerings go to in a ritual. Red Earth makes offerings to the DotO first, and then to other spirits in nine further rounds of offerings. Ocean's Tide makes offerings to all the Kindreds, with one round of offerings for each, but tries to focus the "gods" round on the DotO. Silver Birch mentions the widest variety of praise offerings, though: "The Earth Mother, the Kindreds, DoTO, the fire well and tree and anyone else who happens along."
Jeremy, who didn't specifically mention the DotO, makes offerings to his "Patron, as well as 'the gods' in a more generalized sense, and often to the Lake (which is the major Land Spirit nearby)." It makes for an interseting contrast with Brandon, another solitary, who makes his priaise offerings only to the Deity of the Occasion (but, of course, doesn't make either method wrong).
¹ - Step by Step through A Druid Worship Ceremony - Isaac Bonewits, originally published in The Druids' Progress #4. A key sentence: "It's important to remember that the primary purpose of all these Praise Offerings is to raise as much psychic/magical/spiritual power as possible, all of it focussed around the deities of the occasion. . ."