This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
If you could be transported through shields, they'd be pretty lousy
shields. Just transport a bomb or boarding party over.
Benjamin Chee:
Just a thought here. Says in the TNG Tech Manual that phasers may
be fired one-way through the ship's own shields due to EM
polarization (whatever that means). If this holds true for other
forms of wavicle energy, then one might be able to transport out
one-way through shields, too.
Benjamin points out that in Star Trek III: The Search For Spock,
Klingons transport while their Bird of Prey is cloaked, yet in "The
Search, Part 1" [DS9] the Defiant has to decloak to transport.
Greg Moseley suggests that the differences between the Klingon and
Romulan cloaking devices may be responsible for the discrepancy; the
Defiant cloak is on loan from the Romulans in return for information
about the Gamma Quadrant.
Benjamin adds that in "The Die Is Cast" [DS9] a Romulan ship decloaks
on top of a runabout before it can beam the occupants aboard. But in
"The Way of the Warrior" [DS9] an entire fleet of Klingon ships stays
cloaked until the battle warms up.
And finally, more wisdom from Benjamin:
One more point - the Klingon clunker in Generations had to decloak
before it could beam Soran aboard, didn't it ? We never really were
told why nor do we have much to conjecture from, but this is indeed
an exception to the rule. In a mail to Mike Okuda, he also admitted
that they never really kept track of the cloaks - might have been
coincidence all the way up till Generations.
 
Continue to: