Here is the full list of all 34 Puzzle Together Multiplayer Jigsaw achievements.
Finished a 500+ piece puzzle by yourself
Sent text chat to other players in an open puzzle
Read the guide all the way through
Flipped the box
Completed ten puzzles
Placed at least 25% of the pieces of a large popular open puzzle
Placed the final piece in a large popular open puzzle
Completed a custom puzzle of your own creation
Sorted the pieces
Sorted a selected group of pieces
Moved a selected group of pieces
One year of puzzling
One month of puzzling
Three months of puzzling
One week of puzzling
First puzzle completed
Connected one thousand pieces
Connect one hundred pieces
Completed one hundred puzzles
Completed five hundred puzzles
Completed one thousand puzzles
Completed two thousand puzzles
Completed five thousand puzzles
Completed ten thousand puzzles
Connected 32,768 pieces
Connected one hundred thousand pieces
Connected five hundred thousand pieces
Connected one million pieces
Two years of puzzling
Three years of puzzling
Accepted invitation to join a puzzle
Sent invitation to join a puzzle
Puzzled with 10+ players online at one time
Hosted a puzzle with 20+ players online at one time