  We study the problem of private vector mean estimation in the shuffle model
of privacy where $n$ users each have a unit vector $v^{(i)} \in\mathbb{R}^d$.
We propose a new multi-message protocol that achieves the optimal error using
$\tilde{\mathcal{O}}\left(\min(n\varepsilon^2,d)\right)$ messages per user.
Moreover, we show that any (unbiased) protocol that achieves optimal error
requires each user to send $\Omega(\min(n\varepsilon^2,d)/\log(n))$ messages,
demonstrating the optimality of our message complexity up to logarithmic
factors. Additionally, we study the single-message setting and design a
protocol that achieves mean squared error
$\mathcal{O}(dn^{d/(d+2)}\varepsilon^{-4/(d+2)})$. Moreover, we show that any
single-message protocol must incur mean squared error $\Omega(dn^{d/(d+2)})$,
showing that our protocol is optimal in the standard setting where $\varepsilon
= \Theta(1)$. Finally, we study robustness to malicious users and show that
malicious users can incur large additive error with a single shuffler.
