Each architecture is tested multiple times against different samples of the shared weight

From the paper :

(1) An initial population of minimal neural network topologies is created

(2) each network is evaluated over multiple rollouts, with a different shared weight value assigned at each rollout

(3) networks are ranked according to their performance and complexity

(4) a new population is created by varying the highest ranked network topologies, chosen probabilistically through tournament selection

What is a "shared weight"?

Usually in neural networks, each neuron has their own weights with different values that will be tuned during training.

Shared weight here means that every neuron in the network shares the exact same weight value.

Neither. The architecture performs well independently of weights BUT all weights must be the same e.g. Same network works well when all weights are 5.0 or when all weights are -3.0

Hmm... that collides with what patresh said.

No, it doesn't. Shared weight means all weights are the same value, as I originally mentioned.