Mortgage companies pull an aggregate report called an RMCR that is like a regular credit report but nicely puts all three reports together and it will show 3 credit scores. Typically mortgage brokers will take the middle score when determining eligibility.
You will see inquiries from another company on your credit report. However, the FICO scoring model should take these inquiries into account when shopping around so it won't have the same effect as if you were to apply multiple times for credit over a longer period such as few months. If you request a credit inquiry, wait say three months and then do another one, then it will have a greater, but not that much, impact on your credit report.
That being said, inquiries are not counted as high on your credit report as utilization. When applying for a mortgage, you should have a minimal number of inquiries over the past year. Inquiries will stay on your report for two years but typically only inquiries in the past year will have a significant effect on your credit rating.
Read more here: https://www.mint.com/blog/credit/what-is-a-residential-mortgage-credit-report-072012/