Does any church believe in Sola Scriptura, but without allowing private interpretation? Is there a protestant church which adheres to Sola Scriptura, but considers rightful interpretation to be made by the church only (or elders, synod, etc.)?
The approach to scriptural interpretation taken by the Anabaptist denominations is that of "community hermeneutic".

Community Hermeneutic means that the community of Christians (the church) is responsible and authorized to interpret the scriptures. This doesn't mean the hierarchy of the church, but all members of the church acting together, and attempting to reach consensus.

In practice this means something like:


The pastor of a congregation preaches a sermon, which embodies his understanding of scripture
The congregation hears the sermon. Because of the pastor's training, they respect his opinions. 
The congregation would also meet together in study groups, and each group might come to a slightly different interpretation from the pastor.
The pastor would listen to the interpretations from the groups, and might modify his own understanding as a result. Each group would also hear and respect the interpretations of others.
Representatives of the congregation might also go to a meeting of churches within the denomination, where they presents their church's views, and also allow other churches to inform theirs. They might be called on to form a consensus interpretation for the whole denomination.


No single person, or group of persons, is responsible for forming the interpretation, and no person is excluded from making a contribution to the interpretation. Rather it is the community as a whole that is responsible.

References:


The Anabaptist Network
Graceworks
Mennonite Brethren Herald