Based on a combination of my memory and things I've looked up online. Please correct any mistakes.
Tyne and wear metro (quite a few stations are shared with national rail)
Central Station/Newcastle: gating arrangements are completely seperate, Metro is fully Gated, NR station is only partially gated.
Sunderland: platforms shared by NR and Metro, open station with no gateline.
Heworth: NR side is ungated. I think direct access from the street to the metro station is gated, but the interchange passageway from NR to metro is not.
Manors: two nearby but seperate stations, not sure if either of them is gated.
Navigation road: Tram and rail platforms are on opposite sides of the dual track formation but there is no footbridge so the rail and metrolink platforms have to be accessed seperately from the street. Neither side is gated.
Altrincham: Four platform ex-nr station with two platforms used by metrolink and two by NR. Station is ungated.
Manchester Airport: Ticket gates were recently installed for the NR side, but i'm not sure exactly how they work with resepct to the metrolink, from the NR map it looks like one metrolink platform is outside the gated area but one is inside it. Perhaps the second metrolink platform is rarely used.
East Didsbury: two entirely seperate stations some distance apart, neither is gated.
Eccles: two seperate stations, I don't think either is gated.
Manchester Picadilly, NR side has partial/leaky gating arrangements (discussed above). Metrolink side is (like all metrolink stations) ungated.
Deansgate: Stations on opposite sides of the road with an interchange bridge, neither side is gated.
Victoria: NR side is gated, metrolink side is open.
Rochdale: seperate, though nearby stations. NR station is gated, Metrolink station is open.
Ashton under lyne: seperate stations a fair distance apart, neither appears to be gated.
Merseyrail (national rail)
(defining merseyrail as "merseyrail electrics")
Southport - shared gateline
Ormskirk - appears to be ungated.
Headbolt lane - appears to be ungated
Bidston - platforms shared between merseyrail and regular rail services, appears to be ungated
Liverpool lime street - On the mainline side, based on the information I can find online, I think 1-5 have barriers while 6-10 are open, and I'm pretty sure the low-level merseyrail platform has it's own seperate barriers.
Liverpool South parkway - seperate ticket gates for the mainline and merseyrail sides.
Hunts cross - appears to be ungated.
Ellsmere port - appears to be ungated.
Chester - all platforms behind the same gateline
Meadowhall - both sides appear to be ungated.
Sheffield - both sides are ungated.
Rotherham Central - tram platforms are accessed via ramps from the end of the rail platforms (trams use the same tracks as trains but their boarding height is lower, so they can't use the same platforms), appears to be ungated.