Do diploids such as humans end up with a mixture of two different shaped proteins where the appropriate gene is expressed differently in the two halves of a chromosome pair or is there some governing factor that dictates that only one of the alleles is ever used to direct protein synthesis?
Yes, that mixture of shapes can happen. Differences in the nucleotide base sequences between the two alleles of a gene can lead to differences in amino acid sequence, which in some cases large differences in protein shape. One example would be if one allele changed a cystine codon to any other codon, leading to loss of a disulfide bond needed to stabilize the tertiary structure of a protein. For an extreme example, consider a mutation causing sickle-cell anemia; in this case, the disease-causing allele leads to altered shape of the entire red blood cell.
If a heterozygote carries a healthy allele and a disease-causing recessive allele, the protein produced by the healthy allele provides enough activity to mask the lack of activity of the "broken" protein provided by the recessive allele. If, instead of being a true recessive, the disease-causing allele is partially penetrant, then the heterozygote would suffer from the reduced level of the active protein (only one allele is providing the good stuff).