I'd have to say that it's possible, under the right circumstances.
Episode 24 of season 6 shows that Commander William Riker was accidentally duplicated in an incident described as a transporter accident. While being transported back to his ship, part of the transporter beam was reflected back to the planet, where Riker was re-materialized as well as being re-materialized on the ship.
Using a break in the disruption field, the away team beams down and discovers a man who appears exactly like Riker. He says that he is Riker, and has been living alone on the base for eight years ever since the Potemkin was unable to transport him back aboard, under the assumption the Potemkin crew presumed him lost. He is a Starfleet Lieutenant, Senior Grade. This was Commander Riker's rank before he was promoted as a result of this mission.
Returning to the Enterprise, Dr. Crusher determines that this person is truly a second Riker; Chief Engineer La Forge postulates that years before, when Riker was being beamed off the planet, the Potemkin had split the transporter beam to cut through the distortions, but one beam was reflected back to the base, so that Riker materialized in both places.
Riker spent 8 years on-board ships with friends who would know him to be "different", if that would have happened. During the episode, people remarked on how his "clone" from the planet was so remarkably like the "real" Riker, including Riker himself. They remembered shared memories and thought similarly, up to the point where the 8 years in different environments caused them to think differently due to having different experiences.
Because Riker was duplicated successfully, I believe Commander Data, and his robotic family, could be similarly duplicated, if someone really tried.
Replicator vs. Transporter technology
I don't remember where I first heard it, but replicators and transporters share a technological family tree. I could swear I'd heard it within one of the TV series (like Commander LaForge mentioning it), since I've never read a novel, but the Wiki page seems to back this up, but from a non-canon source.
According to an academic thesis: "The so-called 'replicators' can reconstitute matter and produce everything that is needed out of pure energy, no matter whether food, medicaments, or spare parts are required." A replicator can create any inanimate matter, as long as the desired molecular structure is on file, but it cannot create antimatter, dilithium, latinum, or a living organism of any kind; in the case of living organisms, non-canon works such as the Star Trek: the Next Generation Technical Manual state that, though the replicators use a form of transporter technology, it's at such a low resolution that creating living tissue is a physical impossibility.
If this were true, in the Star Trek universe, then we'd either need to just give replicators more resolution, or just re-purpose transporters to be replicators. There may be a political issue with this, which may be why replicators are low-res and may be off topic of the TV, films, and even books to not mention it. If you are able to duplicate Riker accidentally, then human and other cloning becomes much more of a possibility and scares people even more about their use. (Consider real life as well as the movie "The Prestige".)
Existing example vs programming
The transporter does work with an existing example of the thing it reconstitutes, but there's also the pattern buffer. People have been "stuck" in the pattern buffer and re-materialized. There have also been people "lost" within that pattern buffer. The question really becomes a question of storage, I think.
For a replicator, you can program things from a toy for a child to an edible pudding. You can also program in repair parts for engineering or nearly anything else. Technically speaking, these are simply patterns that have been reduced to their most basic examples and probably use common subroutines/methods of reproducing the item. Some things are a "simple" compound of a single molecule, but other items include much more complicated combinations. Asking for a sandwich may include bread, lettuce, ketchup, mayo, mustard, meat, a plate, a cup, and a drink. All come out at the same time, so there's plenty of complexity to deal with there.
If an engineer were able to hold onto a pattern buffer with an object in it for long enough to find a way to reduce the amount of data necessary to replicate "compile" it down to a much smaller dataset, it could be possible to store that pattern for later use. I remember Beverly Crusher (or maybe another ships Doctor) say something about checking the DNA in the logs for the last times a person went through the transporter to see if a marker or something similar (maybe a disease) could be used as a anti-virus. The DNA sequence, of course, is just a subset of the data necessary to rebuild a person, so that would be easier to store than the whole transporter pattern buffer, but if they can pull that out and store it for every use of the transporter, there's got to be a way to do it for more data. (No pun intended, but I'll take it.)
As always, the longer technology is around, the more likely it'll get easier to use, the smaller it usually gets, and the more reliable it tends to be. The Star Trek we've seen so far may simply not have progressed far enough to technologically be able to do it yet.
Parallel track from another Question
It also seems that this has been researched and discussed in a different manner. This question doesn't ask about Data being duplicated, but rather the transporter pattern buffer logs being used to restore the dead from their last transportation sequence.
Why can't a Federation Starship's transporter history be used to bring someone back to life?