The Barratt-Eccles operad is an operad in simplicial sets
that provides a particularly nice model of an E_{∞}-operad;
algebras in spaces over the Barratt-Eccles operad model E_{∞}-spaces,
i.e., homotopy coherent commutative monoids in spaces.
It can be described concretely by applying the nerve functor
componentwise to an operad in groupoids, which itself is obtained by
applying the codiscrete groupoid functor componentwise
to an operad Σ in sets such that Σ(n) is the symmetric group of order n
and the operadic composition Σ(n)×(Σ(a₁)×⋯×Σ(aₙ))→Σ(a₁+⋯+aₙ)
is given by stacking the permutations in Σ(aᵢ) together and composing them
with the block permutation in Σ(a₁+⋯+aₙ) induced by the permutation in Σ(n).
Here the codiscrete groupoid functor sends
a set X to the groupoid with X as the set of objects
and exactly one morphism between any pair of objects;
it is the right adjoint to the forgetful functor
from groupoids to sets that sends a groupoid to its
underlying set of objects.

I am interested in similarly spirited constructions
for various cousins of E_{∞}-spaces.

Specifically, I am interested in
group-like E_{∞}-spaces, which can be thought of as homotopy coherent commutative groups and are a model for connective spectra.

Another interesting case is E_{∞}-ring spaces,
which can be thought of as homotopy coherent commutative rings,
and are a model for connective E_{∞}-ring spectra.

As pointed out by Peter May in his answer, operads cannot model such structures because they do not allow for operations with multiple outputs, e.g., diagonal maps, so a part of the question is what type of structure one should use.

For example, simplicial algebraic theories (see http://ncatlab.org/nlab/show/(%E2%88%9E,1)-algebraic+theory) seem to be a viable option. In particular, the Barratt-Eccles construction admits a particularly elegant formulation in terms of a (2,1)-algebraic theory (i.e., a groupoid-valued algebraic theory), see http://ncatlab.org/nlab/show/(2,1)-algebraic+theory+of+E-infinity+algebras.

**Is there an analog of the Barratt-Eccles construction for group-like E _{∞}-spaces and E_{∞}-ring spaces?**