If each router is running either just OSPF or RIP, they cannot share information. However, if one of those routers is running both IGPs, it can, as noted by Jon, redistribute route information from one IGP to the other. Such redistribution can be one way or both ways, and might contain all the information known to one IGP sent to the other IGP or just a subset of that information. When passing information between IGPs, you may run into "translation" issues, as different IGPs often don't carry exactly the same attributes. For example, RIP has "distance" to a destination based on router hops (contained by 4 bits) while OSPF has a sum of link costs to the destination (contained by 16 bits). Or, RIPv1 works with classful addresses while OSPF supports classless addresses. (Also when doing two way redistribution, if there's more than one redistribution router, you generally want to avoid feeding back the route information received from the other IGP back to it. I.e. you need to be extra careful what's shared.)
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Oh, and if you do have more than one IGP on the same router, although, again by default, the two IGPs don't share any information, the router, itself, generally builds a route table using information from all the active IGPs on it. If there's overlapping route information, the router has "rules" on which IGP is considered to have "better" information. By default, OSPF route information is considered better than RIP route information, to the same destination.
