If the distribution of sample means tends to Gaussian, then the distribution of sample sums tend to the same Gaussian (except scaled.)
So if the original X_i are drawn from any distribution with defined with mean m and variance s², then the CLT says that in the limit, the sample mean is distributed N(m, s), and, equivalently, the sample sum is distributed N(nm, sqrt(n)s).
The question is when the "in the limit" results starts being practically useful, and that depends on the distribution of the original X_i.
So if the original X_i are drawn from any distribution with defined with mean m and variance s², then the CLT says that in the limit, the sample mean is distributed N(m, s), and, equivalently, the sample sum is distributed N(nm, sqrt(n)s).
The question is when the "in the limit" results starts being practically useful, and that depends on the distribution of the original X_i.