Haskell, 28 bytes

f(x:y:z)=x==y&&f z f z=z==[] 

Try it online!

Very straightforward. Double speak is only empty or a repeated character prepended to double speak.

Haskell, 28 23 bytes

f(x:y:z)|x==y=f z f[]=1 

Try it online!

Very straightforward. Double speak is only empty or a repeated character prepended to double speak.

Less straightforward now. Outputs via presence or absence of an error, per meta consensus; no error means double speak. Pattern matching fails when the first two characters differ or when there are an odd number of characters. Thanks to Laikoni for these savings!