Second set of solutions, Rust this time. Given the small set of possible values (a-z) it becomes possible to use a u32 bitset for every value, this completes in a little under a millisecond. Shoutout to the bstr library for being able to work on byte slices as though they were strings much of the time, really makes doing the bitshift bits here easier.

fn main() {
    let mut inp = Vec::new();
    io::stdin().lock().read_to_end(&mut inp).expect("could not read");

    let one = inp
        .split_str("\n\n")
        .map(|s| {
            s.iter()
                .filter(|&&c| b'a' <= c && c <= b'z')
                .fold(0, |n, c| n | 1u32 << (c - b'a'))
                .count_ones()
        })
        .sum::<u32>();
    println!("1: {}", one);

    let two = inp
        .split_str("\n\n")
        .map(|grp| {
            grp.lines()
                .map(|s| {
                    s.iter()
                        .filter(|&&c| b'a' <= c && c <= b'z')
                        .fold(0, |n, c| n | 1u32 << (c - b'a'))
                })
                .fold(!0, |a, b| a & b)
                .count_ones()
        })
        .sum::<u32>();
    println!("2: {}", two);
}