Dustin, perhaps investigate "pop up drainage emitter". One near the curb gets rid of the need to cut the curb & permits & bond posting.
How It Works:
You install your piping from downspout to to where you want the water to exit; in your case next to the curb. The pop-up is on the end and looks like a flat, green disc on the lawn. When the pipe is full of water, the disc pops up and the water escapes, rolling into the street.
They come made for ABS, PVC, NPT and corrugated tubing- in various pipe sizes. They come in Schedule 30 and 40. The cost is about $20.00.
Some have drip legs so any roof debris that arrives will settle in the drip leg. You pull the cap off and reach in to take out the compost.
In a house I had, I had the roof downspouts connected to buried pipe that ended perhaps 50 feet away in a pop-up disc. After a rain when the pipe had enough water in it, the disc would pop up and the water would empty. You could hear the vacuum sound in the downspouts. On light rains, water would seep out around the disc, perhaps lifting it a little bit.
It worked great and the mower went right over the disc. I'll never now why, but it worked in winter when the ground was frozen far deeper than the pipe. When connected to sump pumps, they spray like crazy and are fun to watch.
Tunnel Time! (How to get under the sidewalk)
To get conduits under rather narrow (perhaps one or two lanes wide) concrete when the shovel guys were busy and I was too lazy (or the job was too far away) to go get a directional boring unit from the shop, I used my "Whack-A-Pipe" method.
I'd aim for an area where I didn't think there were large tree roots, then dig my conduit (rain pipe in your case) trenches until they reached each side of the slab.
From the trench, I would take a piece of rigid conduit or plumbing pipe with a cap on each end and hammer it under the slab until one end appeared on the other side. A mini sledge works well, as does a real sledge. (See * below for cheaper options)
When the pounded-on conduit appeared on the other side, I would remove one cap from the pounded-in length and attach a length of conduit (water pipe in your case) to it with a coupling and pull the pounded on length. If it is clay, dish soap on the second pipe helps when pulling the first pipe.
I'd repeat until the slab was crossed.
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For large diameters to make pounding easier, instead of a cap on the forward end, install a bell-shape reducing coupling and a plug to make sort of a spear point. This worked well in clay.
If it is PVC or ABS, you may be able to pound in your actual pipe instead of using a piece of conduit. If it's large such as 3", place a board across the end and pound the board.
* Cheaper Than Buying Steel Pipe: You can also use rebar or even a 2 x 2 wood if the dirt cooperates. Use a worm clamp to attach the plumbing pipe to the thing being pounded upon.
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The Messy But Fun Way:
If yours is sand under the slab, attach your drain pipe near the end of a garden hose. Tape the end of the plumbing pipe closed. Put a pressure nozzle on the garden hose and push the pipe while the nozzle washes away the sand in front of it. When done, pack in new sand and tamp with a board and sledge sideways (to prevent cracking the slab above later).
The Manly-Man Way:
A typical 5 foot x 3 foot x 3 inch sidewalk slab can be lifted by one edge and dragged out of the way so you can dig a trench instead of pounding a pipe under. Two people make it a whole lot easier.
Or you can lift an edge and flip it carefully onto the grass. Be sure to tamp the back-fill very well.
PS: In my neighborhood, people have sump pumps draining through curb cuts. Boy what a giant ice rink they make in winter!
Enjoy Your Project!
Paul