Do weak acid/weak base neutralisation reactions go to completion?
Weak acids and bases do indeed remain in an equilibrium state with a certain concentration of the free acid and the free base alongside a certain concentration of the deprotonated acid anion and the protonated base cation.

In general, you can consider any acid-base reaction to be essentially the following:

$$\ce{HA &lt;=&gt; H+ + A-}\tag{1}$$

Which in turn means that the $\mathrm pK_\mathrm a$ value is defined as in $(2)$:

$$K_\mathrm a = \frac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]}\tag{2}$$

Reactions $(1)$ and $(2)$ are usually written out in water with hydronium in place of a naked proton but the same principle applies. Of course, a base reaction can be considered essentially the reverse, meaning that the acid constant typically given is actually $\mathrm pK_\mathrm a (\ce{HB+})$ rather than anything directly corresponding to the free base. If we now take the reaction of a (weak) acid and a 
(weak) base, we get equation $(3)$ and the equilibrium constant as given in $(4)$:

$$\begin{align}\ce{HA + B &amp;&lt;=&gt; A- + HB+}\tag{3}\\[0.7em]
K &amp;= \frac{[\ce{A-}][\ce{HB+}]}{[\ce{HA}][\ce{B}]}\tag{4}\end{align}$$

We can now perform simple mathematics with $(4)$ to arrive at the modified equation $(5)$ as below:

$$\begin{align}K &amp;= \frac{[\ce{A-}][\ce{HB+}]}{[\ce{HA}][\ce{B}]}\tag{4}\\[0.7em]
&amp;= \frac{[\ce{A-}][\ce{HB+}][\ce{H+}]}{[\ce{HA}][\ce{B}][\ce{H+}]}\\[0.7em]
&amp;= \frac{[\ce{H+}][\ce{A-}]}{[\ce{HA}]} \times \frac{[\ce{HB+}]}{[\ce{B}][\ce{H+}]}\\[0.7em]
K &amp;= \frac{K_\mathrm a(\ce{A})}{K_\mathrm a (\ce{HB+})}\tag{5}\end{align}$$

We have thus arrived at a way to determine the equilibrium constant of the proton transfer reaction just from the acidity constants of both participants. If both acids weak and have similar $\mathrm pK_\mathrm a$ values — as ammonium and acetic acid do — the equilibrium constant will be close to $1$ and thus both sides of the equation will have similar concentrations.

Using $(5)$, you can also predict a ‘degree of completion’ for any acid-base reaction.