Gap-Hamming with different "threshold" (i.e., not $n/2$)
As it turns out, the one-sided error case (public coin) was addressed in [KK16], which shows the communication complexity is then $\tilde{\Theta}\!\left(\frac{(t-g)^2}{t+g}\right)$ (the upper and lower bound are off only by a $\log(t-g)$ factor).

(so, for $\mathsf{GHD}_{n,2\sqrt{n},\sqrt{n}}$, the one-sided public-coin communication complexity is $\tilde{\Theta}\!\left(\sqrt{n}\right)$.)

[KK16] One-sided error communication complexity of Gap Hamming Distance, Egor Klenin and Alexander Kozachinsky. ECCC TR16-173, 2016.