Brachylog, 17 bytes

c@~P,?:{l~l}a.cP,

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###Explanation

We basically create a list of sublists with variable elements that has the same "shape" as the Input, and then state that if we concatenate everything into a single list, it must result in a shuffle of the concatenation of the input into a single list.

c@~P,                 Concatenate the Input into a single list. Shuffle it and call that P.
     ?:{   }a.        The Output is the result of applying this to each element of the input:
        l~l               The Output is a list of same length as the Input.    
             .cP,     P is the concatenation of the sublists of the Output.

Brachylog, 17 bytes

c@~P,?:{l~l}a.cP,

Try it online!

Explanation

We basically create a list of sublists with variable elements that has the same "shape" as the Input, and then state that if we concatenate everything into a single list, it must result in a shuffle of the concatenation of the input into a single list.

c@~P,                 Concatenate the Input into a single list. Shuffle it and call that P.
     ?:{   }a.        The Output is the result of applying this to each element of the input:
        l~l               The Output is a list of same length as the Input.    
             .cP,     P is the concatenation of the sublists of the Output.