- Distance between of your nodes is equal to default value
node distance between right and left side borders of nodes, i.e. between (centeral node.east) and (right node.west).
- For convenience of writing of the distance between their centers I increase this distance to
22mm
- Distance is calculated on the similar way as in @Ignasi answer (+1) with help of the
calc TikZ library:
\documentclass[tikz, border=2mm]{standalone}
\usetikzlibrary{calc,
positioning,
quotes}
\begin{document}
\begin{tikzpicture}[
node distance = 22mm,
place/.style = {circle, draw=blue!50,thick, fill=blue!20,
inner sep=0pt,minimum size=6mm},
transition/.style = {draw, thick, fill=black!20,
inner sep=0pt,minimum size=4mm},
every edge quotes/.style = {auto, font=\scriptsize, inner sep=1pt}
]
\node[place] (centeral node) {};
\node[transition] (right node) [right=of centeral node] {};
%
\draw[|-|]
let \p1 = ($(right node.center) - (centeral node.center)$) in
(centeral node.center) %
to ["\x1",
"\pgfmathparse{scalar(\x1*0.35145980pt)} \pgfmathresult,mm" '] % 1pt ~ 0.35145980mm
(right node.center);
\end{tikzpicture}
\end{document}

Addendum;
Using default value for node distance which is 10mm (or 1 cm if you prefer), decreasing edge labels font size and and increasing its inner sep for better readable of measured values, adding 10mm long red line between nodes, to show default value of the node distance, and set outer sep=0pt that it not influence to nodes distance, the above MWE become:
\documentclass[tikz, border=2mm]{standalone}
\usetikzlibrary{calc,
positioning,
quotes}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture}[
place/.style = {circle, draw=blue!50,thick, fill=blue!20,
inner sep=0pt, outer sep=0pt, minimum size=6mm},
transition/.style = {draw, thick, fill=black!20,
inner sep=0pt, outer sep=0pt, minimum size=4mm},
every edge quotes/.style = {auto, font=\tiny, inner sep=5pt}
]
\node[place] (centeral node) {};
\node[transition] (right node) [right=of centeral node] {};
%
\draw[|-|]
let \p1 = ($(right node.center) - (centeral node.center)$) in
(centeral node.center) %
to ["\x1",
"\pgfmathparse{scalar(\x1*0.351459803pt)}\pgfmathresult,mm" '] % 1pt = 0.35145980mm
(right node.center);
\draw[red] (centeral node.east) -- ++ (10mm,0);% for show default distance between nodes
\end{tikzpicture}
\end{document}
It yields to:

Value 14,9998 mm ~ 15mm is equal to:
<default node distance> + (1/2)<minimum size of left node (=3mm)>+ (1/2)<minimum size of right node (=2mm)>
letpath operation (sorry, commuting now, I can't try an answer). But this question smells of an XY problem... For what do you need the distance? – Rmano Jul 22 '20 at 07:00on gridoption from thepositioninglibrary (p.242, pgfmanual, v3.1.5.b) then useright=2cm of centeral nodeto put the center of your new node at exactly 2cm ofcenteral nodecenter. – Paul Gaborit Jul 22 '20 at 08:11