The dot notation means to write a sum of three terms by matching x, y, z matrices for the two particles, as if $\vec{\sigma}^1$ were a vector $(\sigma_x^1,\sigma_y^1,\sigma_z^1)$ and likewise for particle #2.
$$
\vec{\sigma}^1\cdot\vec{\sigma}^2 = \sigma_x^1\sigma_x^2 + \sigma_y^1\sigma_y^2 + \sigma_z^1\sigma_z^2
$$
Note that $\sigma_i^1$ and $\sigma_i^2$ act separately as matrices; they don't multiply with each other.
We have
$$
\sigma_z^1 = \left(\begin{matrix}1&0\\0&-1\end{matrix}\right)
$$
but acting only on the particle #1 factor of the wavefunction, and
$$
\sigma_z^2 = \left(\begin{matrix}1&0\\0&-1\end{matrix}\right)
$$
action only on the particle #2 factor.
For example, if you have some wavefunction (not necessarily an eigenstate of anything) being particle #1 spin up and particle #2 down,
$$
\psi =\psi^1\psi^2 = {\left(\begin{matrix}1\\0\end{matrix}\right)}^1{\left(\begin{matrix}0\\1\end{matrix}\right)}^2
$$
and some arbitrary matrix acting on particle #1,
$$
U^1 = \left(\begin{matrix}a&b\\c&d\end{matrix}\right)
$$
and another arbitrary matrix for particle #2,
$$
V^2 = \left(\begin{matrix}e&f\\g&h\end{matrix}\right)
$$
then
$$
U^1 \psi = (U^1\psi^1 )\psi^2 = {\left(\begin{matrix}a\\c\end{matrix}\right)}^1{\left(\begin{matrix}0\\1\end{matrix}\right)}^2
$$
and
$$
V^2 \psi = \psi^1(V^2\psi^2 ) = {\left(\begin{matrix}0\\1\end{matrix}\right)}^1{\left(\begin{matrix}f\\h\end{matrix}\right)}^2
$$
Note that tacking superscripts onto column spinors like that to indicate particle identity isn't normal practice outside of textbooks or places where things need to be explained in such detail. A common way to deal with this is to form a four dimensional space, the product of the two spin spaces of the particles. The basis vectors would be $\uparrow^1\uparrow^2, \uparrow^1\downarrow^2, \downarrow^1\uparrow^2, \downarrow^1\downarrow^2$ or some nice linear combination.
Think about it: we need an index to count along the spinor components (the 2-dimensional Hilbert space of complex numbers), an index to count spatial dimensions (x,y,z), and an index to count particles. We're dealing with a three-dimensional entity, a matrix with rows, columns and "another". Stick with theoretical physics, and we'll add chromodynamic charge, more spatial dimensions to deal with curved spacetime, and sprinkle in a dash of supersymmetry or technicolor or whatever is hot with the kids these days. And you'll refer to the Pauli matrices as "Clebsch-Gordan coefficients" but at least they're the simplest nontrivial case. Have fun!