How do you label an equation with something on the left and on the right?

This could be an overkill. Yet you only need to say \rtag{stuff} to get the a tag on the right. Note also that amsmath makes some efforts that the tag does not overlap with the equation. This solution does not make such efforts.

\documentclass{article}
\usepackage[leqno]{amsmath}
\usepackage{tikzpagenodes}
\newcommand{\rtag}[1]{\begin{tikzpicture}[baseline=(tmp.base),remember picture]
\node[inner sep=0pt](tmp){\vphantom{1}};
\begin{scope}[overlay]
 \path (current page text area.east|-tmp.base)
  node[anchor=base east,inner sep=0pt,outer sep=0pt]{(#1)};
\end{scope}
\end{tikzpicture}}
\begin{document}
\begin{equation}
 \sum\limits_{n=0}^{\infty} x^n = \frac{1}{1-x} \rtag{$|x|<1$}
\end{equation}

\begin{equation}
  \rtag{$|y|<1$}2\sum\limits_{n=0}^{\infty} y^n = \frac{2}{1-y}
\end{equation}

\end{document}

You have to do some manual intervention since it's not the usual way of doing things:

\documentclass[leqno]{article}

\usepackage{amsmath}

\makeatletter
\newcommand{\ltxlabel}{\ltx@label}
\makeatother

\begin{document}
See \eqref{eqn:sum_x_n_1}.
\[
  \refstepcounter{equation}\ltxlabel{eqn:sum_x_n_1}
  \makebox[\linewidth]{%
    \rlap{(\theequation)}\hfill
    $\displaystyle \sum_{n = 0}^\infty x^n = \frac{1}{1 - x}$\hfill
    \llap{($\lvert x\rvert < 1$)}%
  }
\]
And also see \eqref{eqn:sum_x_n_2}.
\begin{equation}
  \sum_{n = 0}^\infty x^n = \frac{1}{1 - x} \label{eqn:sum_x_n_2}
\end{equation}

\end{document}

A more typical way of managing conditions is via a \quad (or \qquad) separation:

\begin{equation}
  \sum_{n = 0}^\infty x^n = \frac{1}{1 - x} \qquad (\lvert x \rvert < 1)
\end{equation}