Its kind of what Casper is trying to do with its sharding scheme. The high level idea is to split the network into several groups, and a group of nodes will be responsible for verifying some set of transactions (or a shard).
It is entirely possible to do in a private network, where all the nodes are known and have fixed identities. However, in an open network, there are many difficulties. For example, how do we determine which nodes to verify a specific set of transactions (or a shard), how to handle churn (nodes join and leave), how to guarantee that a group is not doing something malicious (e.g. accepting invalid transactions), etc. Its extremely hard to do in Bitcoin (there is no publicly known solution so far), and its many times harder to do in Ethereum with the present of stateful smart contracts.