From this JKR interview (thank you, Dason):
In the fifth book, Harry can see the Thestrals. Can you?
Yes, I can, definitely. That is a really good question, because it enables
me to clear up a point. The letters that I’ve had about the Thestrals!
Everyone has said to me that Harry saw people die before could see the
Thestrals. Just to clear this up once and for all, this was not a mistake.
I would be the first to say that I have made mistakes in the books, but this
was not a mistake. I really thought this one through. Harry did not see his
parents die. He was one year old and in a cot at the time. Although you never
see that scene, I wrote it and then cut it. He didn’t see it; he was too
young to appreciate it. When you find out about the Thestrals, you find that
you can see them only when you really understand death in a broader sense,
when you really know what it means. Someone said that Harry saw Quirrell die,
but that is not true. He was unconscious when Quirrell died, in Philosopher’s
Stone. He did not know until he came around that Quirrell had died when
Voldemort left his body. Then you have Cedric. With Cedric, fair point. Harry
had just seen Cedric die when he got back into the carriages to go back to
Hogsmeade station. I thought about that at the end of Goblet, because I have
known from the word go what was drawing the carriages. From Chamber of Secrets,
in which there are carriages drawn by invisible things, I have known what was
there. I decided that it would be an odd thing to do right at the end of a
book. Anyone who has suffered a bereavement knows that there is the immediate
shock but that it takes a little while to appreciate fully that you will never
see that person again. Until that had happened, I did not think that Harry
could see the Thestrals. That means that when he goes back, he saw these spooky
things. It set the tone for Phoenix, which is a much darker book.
I believe that seeing someone pass through the veil could allow that person to see the thestrals, if they "understand death in a broader sense" and/or "appreciate that they will never see that person again".