What are the functionality, security and privacy implication related to my choice of Ethereum nodes?
Functionality will be pretty much the same either you choose to use your own node or a public node. But when it comes to privacy and censorship resistance using a local node is the best choice.
However, using a local node carries an increased load for maintaining synchronized a node connected to the Ethereum network. Since light nodes can be used for this task, its not a problem in terms of computational power or storage (Unless a user wants to use a full node, then storage becomes an issue). Additionally, when running your own node security can be an issue, specially if your transaction volume is high. There will be people scanning ports, looking for vulnerabilities. The risk is not extreme but additional precautions must be taken regarding ports and firewalls. So with a remote node you delegate security to the node provider but with a local node you get to choose your own cybersecurity like choosing your own firewall, using a VPN, picking geographic location / servers, etc.
On the other hand, exclusive use of public nodes like Infura can lead to centralization of resources and possible censorship. Don't get me wrong, Infura, metamask and others provide a valuable product for people, which makes UX more seamless and they do a great job. But if there were pressure from a party to censor a particular dapp or a particular market in the case of Augur, companies in charge of running nodes like Infura could, in theory, blacklist certain content leaving no other choice than running a local node or looking for another node provider to connect to said dapp or market.
As a personal opinion it depends on where on the spectrum of ease of use vs. censorship resistance a user needs lay. Right now we aren't there just yet, we are in a building phase but censorship pressures will come and there must be alternatives for all types of users from all over the world regardless of local censorship policies.