In a copper wire, how many electrons does each atom give away?
This is a number which can be measured via the Hall effect. This reference give the Hall coefficient as $-5.4\times10^{-11}\,\mathrm{m^3/C}$ for a number density of charge carriers as
$$n_\mathrm e=\frac1{\left(-5.4\times10^{-11}\,\mathrm{m^3/C}\right)\left(-1.602\times10^{-19}\,\mathrm C\right)}=1.16\times10^{29}/\mathrm m^3$$
The number density of copper ions is
$$n_\ce{Cu}=8920\,\frac{\mathrm{kg}}{\mathrm{m^3}}\times\frac{1000\,\mathrm g}{\mathrm{kg}}\times\frac{1\,\mathrm{mol}}{63.546\,\mathrm g}\times\frac{6.022\times10^{23}}{\mathrm{mol}}=8.45\times10^{28}/\mathrm m^3$$
So that works out to about $1.37$ charge carriers per ion.