Anna doesn't regain her memories during the movie. Frozen's writer/director Jennifer Lee has confirmed this on Twitter:
Since "Elsa filled her in," Anna would know that her memory of skating with Elsa was a false one, even though she didn't have her old memories back.
Additionally, a follow-up book, Memory and Magic, is going to deal with Anna trying to get back her lost memories:
After years apart, Anna and Elsa are finally getting to know each other as sisters. But Anna still wishes she could remember the magical times they had together when they were younger--skating and making snowmen, even in the middle of summer! An eager young troll claims he can restore the memories that Pabbie removed, but Elsa's not so sure. Maybe it's time to make new memories!
Whether she'll regain her lost memories in the upcoming book remains to be seen.
Update: Memory and Magic is out now. It confirms that Anna doesn't get her memories back. A troll named Brock gives her a potion and tries to use a spell to restore them, but he isn't able to.**
But Anna was still hopeful. Just because Brocks magic looked different
didn't mean it didn't work. "Now I'm going to ask you some questions,
Just Anna," Brock said. "The answers will prove that my magic has
worked!" Anna nodded. Elsa looked at Brock expectantly. "First
question: what color is the sky?" Brock asked. "Blue," Anna answered.
The troll looked very pleased. "What color is your hair?" he asked.
"Red," Anna replied easily. "What does this have to do with anything?"
Kristoff grumbled. "I beg you not to interrupt, Book Crusher," Brock
said. Kristoff opened his mouth to argue, but Elsa silenced him with a
look. Brock looked eagerly at Anna. "What is your favorite food?"
"Chocolate," Anna said. "Aha!" Brock exclaimed. "Success! Your memory
is saved, Just Anna!" "Brock, you offered to restore my memories of
Elsa's magic! From our childhood! None of those questions had anything
to do with magic!" Anna said. Brock the Mystical scratched his chin
thoughtfully. "Oh, right," he said. "I knew I forgot something. Maybe
you should cluck some more." Anna's mind clouded with disappointment.
Brock wasn't mystical. He certainly wasn't going to be powerful like
Grand Pabbie. He was nothing more than a kooky little troll who liked
to brew awful potions. Anna's shoulders slumped. She glanced at Elsa
and Kristoff, expecting them to say "I told you so." But they both
looked just as disappointed as she felt. Anna realized suddenly that
they both had been hoping it would work, too. They really cared about
her. It made Anna feel just a little bit better.
Anna decides she can accept not getting her real memories back because she has Elsa to tell her what really happened.
After the smoke cleared, the trolls gave a hearty cheer. They were
relieved that Queen Elsa had stopped the fire and that no one had been
hurt. The forest was safe once again. The trolls invited Brock to stay
with them until his hut was rebuilt. Anna, Elsa, and Kristoff helped
him gather his belongings from the rubble. The damage wasn't as bad as
they had first thought. They were even able to save some of his moldy
books. Brock was still convinced that his potion had restored Anna's
memories. Kristoff was eager to tell him the truth, but Anna didn't
have the heart. Brock had tried his best to help her. The troll
deserved to remember as much magic as he wanted to. That night at the
castle, Anna and Elsa sat down to dinner. Anna was excited to recall
the day's events. "I can't believe I actually clucked like a chicken!"
she said. "I can't believe you actually drank that awful potion," Elsa
laughed. "The smell was horrible!" Anna thought about how determined
she had been to get her memories back. It had been important to her to
remember on her own. Only that morning, she had felt like a puzzle
with pieces missing. But this afternoon, all that had changed. Anna
realized that her memories weren't missing. As long as Elsa was there,
the puzzle was complete. Elsa could remember the magic and Anna could
remember the laughter. Together, they were a winning combination.