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