I am seeking to find something that allows me to "broadcast" macros outside a group. Concrete examples include paths and scopes in tizpictures. Here is an M(N)WE.

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

usetikzlibrary{calc}

makeatletter

letsmuggleoutonepgfmath@smuggleone

makeatother

begin{document}

begin{tikzpicture}[globalize/.code n args={2}{xdef#2{#1}},

localize/.code n args={2}{pgfmathsetmacro{#2}{#1}typeout{#2}

%smuggleoutone#1

}]

begin{scope}[local bounding box=extra]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

pgfextra{xdefmyangle{n1}};

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (extra.north) {using verb|pgfextra|};

%

begin{scope}[local bounding box=globalize,xshift=3cm]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

[globalize={n1}{myangle}];

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (globalize.north) {using texttt{globalize}};

%

xdefmyangle{7}

begin{scope}[local bounding box=localize,xshift=6cm]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

[localize={n1}{myangle}];

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (localize.north) {attempt to smuggle};

%

end{tikzpicture}

end{document}

The two options on the left do partly what I am seeking to do, namely broadcast the macro myangle outside the path. However, they do it at the expense of making myangle global. TikZ has some internal commands that may allow one to avoid this, and to just smuggle the macro outside the path. Specifically, @DavidCarlisle suggested in the chat to use pgfmath@smuggleone. However, my above attempts failed, i.e. if I uncomment

%smuggleoutone#1

the code produces errors.

QUESTION: Can one smuggle the macro outside the group without making it global?

"BONUS": Of course it would be great if there was an explanation what all the smuggle commands do.

"BONUUUUS": Conceivably these methods may be useful independently of TikZ, so if there is a way not to make them depend on TikZ being loaded, this would be great, but is certainly not a requirement.

You can "smuggle" definitions out of their group with the TeX primitive aftergroup. I'll first explain what aftergroup does, then give one possible definition of smuggleoutone and finally mention why you might want to use AfterGroup (from etextools) instead.

About aftergroup:

It is possible to insert a single token right after the end of the current group using aftergroup<token>.
You can only smuggle out one token at a time, so if you want to move out something consisting of multiple tokens (like a definition) you'll need to aftergroup each of these tokens separately. This includes things like braces ({}), so for instance

{aftergroupdefaftergroupabcaftergroup{aftergroup Aaftergroup Baftergroup Caftergroup}}

is equivalent to {}defabc{ABC}.

This is quite a hassle, so the following may be more practical:

{gdefsomethingunique{defabc{ABC}}aftergroupsomethingunique}

This works by globally assigning defabc{ABC} to somethingunique and inserting that after the end of the group.
If ABC is replaced by some macro, say ABC, that is only defined within the current group and that you want to be fully expanded then you'll want to use xdef instead:

{%

  newcommand*ABC{ABC}%

  xdefsomethingunique{defnoexpandabc{ABC}}%

  aftergroupsomethingunique

}

I've inserted noexpand in front of abc because we don't want abc to be expanded.
If you only want ABC to be expanded once you can instead use the slightly more complicated

{

  newcommand*ABC{somethingthatshouldntbeexpanded}%

  xdefsomethingunique{defnoexpandabc{unexpandedexpandafter{ABC}}}%

  aftergroupsomethingunique

}

The primitives noexpand, unexpanded and expandafter are all explained in this this answer.

Defining smuggleoutone:

To smuggle the definition of abc out of a group you can do what I did above with ABC replaced by abc itself.
This way abc is defined as itself (expanded once) immediately after the end of the group.

You could thus define your smuggleoutone command as follows:

newcommand*smuggleoutone[1]{%

  xdefsomethingunique{defnoexpand#1{unexpandedexpandafter{#1}}}%

  aftergroupsomethingunique

}

(You meant to use smuggleoutone#2 instead of smuggleoutone#1 by the way.)

There is one tiny problem with this definition though, and that is the fact that somethingunique is no longer unique if you use the same name every time.
It would thus be better to increment a counter every time this command is used and to store the thing you want to smuggle out in a macro whose names contains the value of this counter.
This is how the AfterGroup command, which I describe below, works (except that it also cleans up after itself).

Using AfterGroup instead:

The command AfterGroup is defined in the etextools package.
It acts like aftergroup, but it takes an argument that can consist of any number of tokens.
So, for instance, Aftergroup{defabc{ABC}} inserts defabc{ABC} after the current group without all of the aforementioned hassle.
There's also a starred version, Aftergroup*, that does the same thing but first expands its arguments fully.

An alternative definition would thus be:

usepackage{etextools}

newcommand*smuggleoutone[1]{%

  AfterGroup*{defnoexpand#1{unexpandedexpandafter{#1}}}%

}

This definition shouldn't suffer from the aforementioned problem.

All of this applied to your MWE:

Here is your MWE with two changes: (1) I've redefined smuggleoutone and (2) I've uncommented smuggleoutone#1 and replaced #1 by #2 (I expect this was likely a typo).

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

usetikzlibrary{calc}



usepackage{etextools}

newcommand*smuggleoutone[1]{%

  AfterGroup*{defnoexpand#1{unexpandedexpandafter{#1}}}%

  }

%% Alternative:

% newcommand*smuggleoutone[1]{%

%   xdefsomethingunique{defnoexpand#1{unexpandedexpandafter{#1}}}%

%   aftergroupsomethingunique

% }



begin{document}

begin{tikzpicture}[globalize/.code n args={2}{xdef#2{#1}},

localize/.code n args={2}{pgfmathsetmacro{#2}{#1}typeout{#2}

smuggleoutone#2

}]

begin{scope}[local bounding box=extra]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

pgfextra{xdefmyangle{n1}};

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (extra.north) {using verb|pgfextra|};

%

begin{scope}[local bounding box=globalize,xshift=3cm]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

[globalize={n1}{myangle}];

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (globalize.north) {using texttt{globalize}};

%

xdefmyangle{7}

begin{scope}[local bounding box=localize,xshift=6cm]

path let p1=($(2,1)-(0,0)$),n1={atan2(y1,x1)} in 

[localize={n1}{myangle}];

node at (1,0) {myangle};

end{scope}

node[anchor=south] at (localize.north) {attempt to smuggle};

%

end{tikzpicture}

end{document}