  We introduce two new classes of covering codes in graphs for every positive
integer $r$. These new codes are called local $r$-identifying and local
$r$-locating-dominating codes and they are derived from $r$-identifying and
$r$-locating-dominating codes, respectively. We study the sizes of optimal
local 1-identifying codes in binary hypercubes. We obtain lower and upper
bounds that are asymptotically tight. Together the bounds show that the cost of
changing covering codes into local 1-identifying codes is negligible. For some
small $n$ optimal constructions are obtained. Moreover, the upper bound is
obtained by a linear code construction. Also, we study the densities of optimal
local 1-identifying codes and local 1-locating-dominating codes in the infinite
square grid, the hexagonal grid, the triangular grid, and the king grid. We
prove that seven out of eight of our constructions have optimal densities.
