One (at least I find) elegant and quite usable way would be a new command, say \sidecite.
It will use the same syntax as all the \cite command, but features an optional offset argument enclosed in <> at the beginning.
We will use xparse to implement it.
First we copy the definition of \footcite from the relevant .cbx style file (in your case verbose.cbx), rename it to \sidecitehelper and replace \mkbibfootnote by \bibfootnotewrapper (so we don't actually get a footnote, but consistent output).
\DeclareCiteCommand{\sidecitehelper}[\bibfootnotewrapper]
{\usebibmacro{prenote}%
\toggletrue{blx@footnote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{cite:postnote}}
Then we define \sidecite
\ExplSyntaxOn
\NewDocumentCommand\sidecite{D<>{}O{}om}{%
\iftoggle{blx@footnote}
{\cs_set_protected_nopar:Npn \__sct_wrapper:nn ##1 ##2 {\mkbibparens{##2}}}
{\cs_set_protected_nopar:Npn \__sct_wrapper:nn ##1 ##2 {\sidenote[][##1]{##2}}}
{\IfNoValueTF{#3}
{\__sct_wrapper:nn{#1}{\sidecitehelper[#2]{#4}}}
{\__sct_wrapper:nn{#1}{\sidecitehelper[#2][#3]{#4}}}}
}
\ExplSyntaxOff
With this updated definition of \sidecite we now check if we are in a side note (footnote) and if so only wrap the citation in parentheses instead of creating nested foot/side notes.
We have a dummy command \__sct_wrapper:nn that in footnotes just wraps things in parentheses and outside of footnotes creates a side note.
In biblatex the command \mkbibfootnote takes care of that, and if we hadn't wanted \sidenote to be able to take its optional argument, an approach with a wrapper command mirroring \mkbibfootnote would have been easier; but since we need the optional argument, all this handling has been moved to \sidecite.
There is one problem left with that: tufte-latex's \footnote command can by default not be patched by biblatex, so we need to do that ourselves.
We use xpatch for that
\makeatletter
\xpatchcmd{\@footnotetext}%
{\color@begingroup}
{\color@begingroup\toggletrue{blx@footnote}}
{}
{}
\makeatother
MWE
\documentclass[nobib]{tufte-handout}
\usepackage{xparse}
\usepackage{xpatch}
\usepackage[
style=verbose,
autocite=footnote,
backend=biber
]{biblatex}
\addbibresource{biblatex-examples.bib}
\makeatletter
\xpatchcmd{\@footnotetext}%
{\color@begingroup}
{\color@begingroup\toggletrue{blx@footnote}}
{}
{}
\makeatother
\DeclareCiteCommand{\sidecitehelper}[\bibfootnotewrapper]
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{cite:postnote}}
\ExplSyntaxOn
\NewDocumentCommand\sidecite{D<>{}O{}om}{%
\iftoggle{blx@footnote}
{\cs_set_protected_nopar:Npn \__sct_wrapper:nn ##1 ##2 {\mkbibparens{##2}}}
{\cs_set_protected_nopar:Npn \__sct_wrapper:nn ##1 ##2 {\sidenote[][##1]{##2}}}
{\IfNoValueTF{#3}
{\__sct_wrapper:nn{#1}{\sidecitehelper[#2]{#4}}}
{\__sct_wrapper:nn{#1}{\sidecitehelper[#2][#3]{#4}}}}
}
\ExplSyntaxOff
\begin{document}
This,\sidecite[8]{springer} should\autocite{springer} be a side note and this\sidecite[see][8]{springer} should too.
Lorem ipsum\sidecite<2cm>[9]{geer} dolor\sidecite[cf.][]{springer}.
Sit\sidenote{amet \sidecite{geer}}.
\printbibliography
\end{document}

Note the 2cm offset of footnote 3 (\sidecite<2cm>[9]{geer}), as well as the natural "if there is only one optional argument, it is the postnote" behaviour standard biblatex cite commands in footnote 1 (\sidecite[8]{springer}).
The \sidecite in footnote 6 (\sidenote{amet \sidecite{geer}}) becomes a citation in parentheses to avoid nested footnotes.
\citeand\autocite? – moewe Apr 15 '15 at 08:32