Did you have to accept their Terms and Conditions, and did that include a Certificate (the ones your browser catalogs)?
Note especially that "Private browsing" has no relationship to security of outbound packets. It is related to the computer itself, and only prevents the storing of your browsers visit history, passwords, site cookies, and a few other items. It is not related to 'sent to the Internet ' stuff.
Several important things do need attention
Using Public 'Hotspots' or hotel 'free Internet' general concepts:
Because 'open' networks (those not requiring a password) have no encryption between your unit and the router, everything you send or receive is in plain text.
So a hacker, with a commonly available application, can intercept and log all of your traffic 'packets'; run "brute force" programs to crack passwords and examine all of your traffic at their leisure ("off line").
So the best advice is do not visit sites that are private, or require 'log-in', especially credit cards, banks, on so on.
In theory, websites that use SSL (indicated by the 'https' prefix in the URL) would be safe, but even that has become less than dependable.
A location (with encryption) that uses "gateway" technology can log all traffic, by presenting a "Certificate".
A padlock indicating SSL may in fact be coming from the router (gateway), which may uncloak and record all traffic before re-encrypting and sending it out to the Internet destination.
Always right click the browsers "lock" or site favicon, then examine the pedigree of a certificate's source.
Any goofy or strange authority could mean some monkey business going on.
Also, when you use a "hotspot" you are by definition in that LAN; and some Windows OS settings open certain ports which can be used to access your computer, by the bad guys.
XP-SP3
Turn off (aka 'unbind') printer & files sharing, via:
Start> Control Panel> Network Connections> right click 'Wireless Network Connections> Properties> General tab: here, uncheck "File & Printer sharing..."> OK out.
Vista & Win 7
Win7 and Vista have slightly different configurations for wifi.
There are 2 basic modes: A) Home & Office; or B) Public.
[To access these settings go to Control Panel (All control panel items)> Network and Sharing Center> Change advanced sharing settings]
For "Public" [On the road or unknown connections]
Turn off network discovery (stealth your system)
Turn off file & printer sharing (shields your files from prowlers)
Turn off Public folder sharing (hides your otherwise 'shared' folders)
Turn off Media streaming
Use "128 bit..." (reduces external machine access probability)
Turn on Password protected sharing (adds another layer of security)
After re-configuring any of the above, make sure to "Save changes".
For ALL computers; use a stout firewall.
SUMMATION: If you are not proficient at all this Certificate business, and you use Windows: don't use an unknown Network for high value work.