Storm Sewer Tie-Ins

If you intend to connect to the public storm drain, storm sewer doesn't generally have backflow prevention on gravity systems. You wouldn't have enough pressure to activate passive backflow in the same way you do a potable water system.
I see that you are in BR, so you would want to contact their Transportation and Drainage department to see if they would even allow you to connect to the public storm system. I work for a City in Texas that allows this, but you have to meet some requirements. It's probably not worth the expense and hassle to make an underground connection. Below is an explanation of why, based on what we require.
First - you likely need to provide a set of engineered plans that show the drainage area, calculated discharges, and connection location (plan and profile). The plans would show where and how you expect to tie into the public system. There may be a standard detail if you are punching into a pipe or inlet. You also need to consider if the tie-in location is accessible, and what the subsidiary repairs may require (i.e. is the public facility under a street and would require a road repair?, are there other utilities you may have to cross?) Another consideration is the hydraulic grade. If your home is in a bowl, and the public system is designed to operate at full capacity for a design storm (likely 25-year event), then you could experience surcharge of the public system out of your private system if there isn't sufficient elevation.
Second - Assuming you get through all the design issues, you would need to hire an approved contractor to do the work. Most cities won't let any unqualified, uninsured contractor to touch their facilities without proper bonds and insurance. You may have to have a 3-party contract between you (as the owner), the contractor, and the City to ensure that all responsibilities and obligations are clearly defined.
Third - there is likely an inspection fee from the City to ensure that the improvements are installed in accordance with the plans and specifications. This would probably be a few hundred dollars.

Based on all these requirements, we typically only see commercial developments and new subdivisions tying anything into a public storm system. It is a lot of headache, paperwork, and expense. Your street should be designed to convey any runoff from your property into the closed system. If your house is below the grade of the street, a sump pump is going to be your best route.