Ah, stuff it. Here you go.
Actually it's not a buzzer - as shown, it just illuminates lamps. But that's close enough!
This is a pretty simple circuit using two relays. There are two normally open momentary pushbuttons, SW1 and SW2, for the two contestants, and two lamps (or beepers or whatever you want to use), LP1 and LP2. The other components are a 12V battery or other DC voltage source, two SPDT relays, two 1N4001 diodes, and a normally closed momentary pushbutton that's used to reset the circuit once the winner has been determined.
Normally, both relays are in the state shown - that is, they are both de-energised and have their common contacts connected to their NC (normally closed) contacts. In this state, each relay provides +12V to the opponent's pushbutton.
As soon as one player pushes his button, 12V from his opponent's relay is provided to the coil of his own relay, and his relay closes. When this happens, the NO contact receives +12V and feeds this to the coil, keeping the relay latched ON and the lamp illuminated (or the buzzer buzzing). This also disconnects the +12V supply from his opponent's pushbutton.
The latched relay remains ON until SW3 is pressed, breaking the supply to the circuit. D1 and D2 are required across the relay coils to absorb inductive kickback from the relays, which would cause arcing and contact damage in SW3 when it is pressed to de-energise the latched relay.
You can get pushbuttons with illumination built-in. Some of these use lamps, which can be connected as shown. Some use LEDs for illumination; if they're not designed for 12V powering, these require a series resistor. For an LED current of 20 mA and a supply voltage of 12V you should use series resistors around 470Ω.
You can connect multiple indicators where I've shown the lamps - for example, lamps in the pushbutton, lamps on the wall, buzzers or beepers, and so on. You can also control AC line-powered lamps using suitable interposing relays.
You didn't give your location so I'll assume you're in the U.S. and I'll give you references for Radio Shack and Digi-Key. Radio Shack are convenient but their relay selection is narrow and way overpriced, and their products are mostly of poor quality. Digi-Key have a huge range of good quality components, but their shipping costs can be high. YMMV, as they say.
I've assumed you want a 12V-powered circuit. All the relays I've listed below are 12V types but other voltages are available - 5V, 6V and 24V are all common relay coil voltages. Just change the power supply voltage and the lamp/buzzer voltage to match.
SW1, SW2: normally open, momentary pushbuttons (SPST or SPNO).
Radio Shack's options are not all clearly described; it's possible that some of these pushbuttons are the push-ON, push-OFF (toggle) type, rather than momentary.
Smallish:
http://www.radioshack.com/nte54-385a-switch-round-pushbutton-off-on-spst-3a/55050558.html
A similar one:
http://www.radioshack.com/red-button-spst-pushbutton-switch/2750646.html
One with a black button:
http://www.radioshack.com/3a-125v-spst-normally-open-pushbutton-momentary-switch/2750644.html
Illuminated:
http://www.radioshack.com/spst-125vac-3a-illuminated-pushbutton-switch/2750009.html
They have other options; just search for pushbutton.
Digi-Key have a huge range of options and they clearly define momentary action buttons vs. toggle buttons. Their switches are also likely to be better quality than Radio Shack's, which are all NTE branded. Use this search filter to see their selection. This filter also includes SPST-NC pushbuttons for use as SW3.
http://www.digikey.com/product-search/en?FV=fff40011,fff80061,1080760,1080761,1080765,1080767,1080768,108076a,108076f,1080770,1080772,1080773,1080774,1080776,1080779,108077b,108077d,108077e,108077f,1080781,1080782,1080783,1080784,1080785,1080788,108078a,108078b,1080791,1080792,1080794,1080797,1080799,108079a,108079c,108079e,108079f,10807a0,10807a1,10807a4,10807a6,10807a9,10807ab,10807ac,10807ad,10807b0,10807b1,10807b4,1080808,108080c,108084e,108085c,1080cec,1080e18,114008e,11403bf,2080001,2080007,2080009,2dc0017,2dc0051,2dc0eb4,2dc0eb5,2dc199e,3ac0005,3ac000e,8300017,830001b,830001c,830001f,8300024,8300041,830006c,8300072,14340003,14340013,14340014,1434001f,14340034,14340035,14340036,14340037,14340038,14340039,1434003a,1434003b,1434003c,1434003f,14340041,14340042&ColumnSort=1000011&stock=1&quantity=1&pageSize=250
All of those options look good. There are lots of styles for you to choose from.
SW3: normally closed, momentary pushbuttons (SPNC).
Radio Shack:
http://www.radioshack.com/pushbutton-normally-closed-momentary-switch4-pack/2751548.html
Digikey: See the selection filter for SW1 and SW2, above.
K1, K2: relays, 12V DC coil, SPDT aka SPCO contacts. (DPDT/DPCO relays are also suitable; just don't use the second set of contacts.)
Radio Shack's cheapest suitable option is
http://www.radioshack.com/nte-r53-5d20-12-0a-dc-spdt-12v-relay/55048294.html
Digi-Key have a wide selection. Their cheapest is the TE Connectivity 1461070-5:
http://www.digikey.com/product-detail/en/1461070-5/1461070-5-ND/1427501
Digi-Key also have a relay that responds more quickly (in 0.002 seconds, instead of 0.005 seconds): the TE Connectivity V23026A1002B201:
http://www.digikey.com/product-detail/en/V23026A1002B201/PB283-ND/254500
D1, D2: 1N4001 or 1N4004 or 1N4007 silicon diodes
Radio Shack:
http://www.radioshack.com/rectifier-diodes-25-pack/2761653.html
Digi-Key:
http://www.digikey.com/product-detail/en/1N4001/1N4001FSCT-ND/1532742
LP1, LP2: Lamps, 12V DC. You can use any 12V DC lamp or other device such as a beeper or buzzer, and you can connect several in parallel, as long as the total load current doesn't exceed the rating of the relays or the rating of the power supply.
If you don't use this design, don't worry that I've wasted time with this post. I like to do this, and we get a good Google rank, so someone will find it later and make use of it.
