Hey i have the follwoing code in my document:

\begin{tikzpicture}
\shade[inner color=blue,outer color=white] (3,0) circle (1);

\shade[inner color=blue,outer color=white] (4.8,0) circle (1);
\end{tikzpicture}


As a result i get:

How to make the overlapping shadows from circles look more natural?

I'd use fadings instead of shadings for that.

\documentclass[tikz,border=3.14mm]{standalone}
inner color=transparent!0,
outer color=transparent!100]
\begin{document}
\begin{tikzpicture}
\end{tikzpicture}
\end{document}


EDIT: It might be advantageous to construct the ring in a slightly different fashion that avoids drawing a white disk.

\documentclass[tikz,border=3.14mm]{standalone}