@Blockstack welcome to the forums
The smart contract data resides in your computer and is propagated to all participating clients through the encrypted p2p protocol bitmessage (BitBay Halo). You will have to keep your computer online if you want for your smart contracts to be searchable but the client will not consume any measurable amount of processing power for this process.
There is a new block on the network every 64 seconds and any transaction gets included on the next block.
When you send from any wallet the BAY will arrive on the receiving wallet on the next generated block which is roughly within 1 minute. Any transaction is usually confirmed within 10 blocks so within about 11 minutes the receiving wallet can actually spend it. Thus the the speed of transaction is about incredibly faster compared to bitcoin (at least 10 times faster).
If you are interested in future scaling here is more technical aspect from our core developer:
There are some ways, increase block size, pruning, node reputation system/chain (for pruning and server nodes), powerful incentivized server nodes (similar to dpos but not as centralized), sidechains and shared staking by sharding (looking at address and grouping it to certain txids and nodes partnering up to share workload). And lightning may work well in tandem with frozen coins as collateral but that’s not our primary area of interest for scaling.