If you've got tubes I'd be a little concerned without the screws. If the tires move at all on the bead they can shear the valve stem off the tube. JMO, but I'd put in the screws...
Like Jay and the others said, the screws were used to keep from tearing off the valve stems ( which leads to a quick loss of air or at minimum having the stem rotate into the rim and not be accessible ) if the tire moves on the rim.
A few observations I've made - some brands of slicks have thicker sidewalls, or more rubber sidewall coating, than other brands. Likewise some specific part number slicks have thicker sidewalls. When I used to run Goodyear slicks the sidewalls were very porous and would leak air right out the sidewall ( it can be checked with the old soapy water trick ). I had to run tubes in those and I used bead screws. I've switched to using Hoosier slicks and I've found that so far they seem to hold air without tubes and I've successfully run some of them without screws. That's not to say there is anything wrong with Goodyear slicks, it's just a characteristic of the size and type I used. Some Goodyears, like the radial slick some of the NHRA stock eliminator guys run, have a thick, stiff sidewall and are usually run without a tube. I don't have any first hand experience with M/T slicks, or the other brands, but my racing friends do and they've told me similar things - it just depends.
Tubes can give you a bit of a safety factor when running slicks, and can help keep from having to add air as often, but also add weight, expense ( the proper natural rubber tubes are very expensive ), and the possible need to run bead screws. If you run tubes you should put some talcum powder in the tire also to keep the tube from sticking to the inside of the tire and also chaffing.
It can depend also, both if they will hold air and if they will slip on the rim, on whether you are using brand new fresh slicks or used slicks. It does seem to becoming more common for racers not to run tubes or screws than it used to be.
