I'm sure ` means something totally different than " to the remote shell. The $ in $i also needs to be escaped (or else Perl will try to interpolate its variable).

@data = `ssh $key 'for i in \`ls /home/dist/\` ; do cat \$i ; done'`;
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or

@data = qx!ssh $key 'for i in `ls /home/dist/` ; do cat \$i ; done'!;
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However, isn't
for i in `ls /home/dist/` ; do cat $i ; done
the same as
cat /home/dist/*?
If so, the following would be simpler.

@data = `ssh $key 'cat ls /home/dist/*'`;
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