We study the sample complexity of the prototypical tasks quantum purity estimation and quantum inner product estimation. The former has been a key subroutine for experimental benchmarking and entropy estimation, while the latter is a core task for cross-platform validation that compares the unknown quantum states produced by different physical platforms. In purity estimation, we are to estimate $\tr(\rho^2)$ of an unknown quantum state $\rho$ to additive error $ϵ$. Meanwhile, for quantum inner product estimation, Alice and Bob are to estimate $\tr(\rho\sigma)$ to additive error $ϵ$ given copies of unknown quantum state $\rho$ and $\sigma$ using classical communication and restricted quantum communication. We propose a protocol that solves quantum inner product estimation with k-qubit quantum communication and unentangled local measurements using $O(\med\{1/\epsilon^2,2^{n/2}/\epsilon,2^{n-k}/\epsilon^2\})$ copies of $\rho$ and $\sigma$. Our protocol can be modified to estimate the purity of an unknown quantum state $\rho$ using k-qubit quantum memory with the same sample complexity. We also prove that arbitrary protocols with k-qubit quantum memory that estimate purity to error $ϵ$ require $({1/^2,2{n/2}/,2^{n-k}/2}) $ copies of $\rho$. This indicates the same sample complexity lower bound for quantum inner product estimation with one-way k-qubit quantum communication and classical communication, and unentangled local measurements. We further improve the sample complexity lower bound for any purity estimation protocols using an identical single-copy projection-valued measurement. Last, we investigate a decisional variant of quantum distributed inner product estimation without quantum communication for mixed state and provide a sample complexity lower bound.