We present ALMA J=3{--}2 and VLA J=1{--}0 observations of the dense molecular gas tracers HCN, HCO+, and HNC in two lensed, high-redshift starbursts selected from the Herschel-ATLAS survey: H-ATLAS J090740.0‑004200 (SDP.9, {z}{spec}=1.575) and H-ATLAS J091043.1‑000321 (SDP.11, {z}{spec}=1.786). In SDP.9 we have detected all J=3{--}2 transitions and also HCN(1–0) and HCO+(1–0). In SDP.11 we have detected HCN(3–2) and HCO+(3–2). The amplification factors for both galaxies have been determined from subarcsecond-resolution CO and dust emission observations carried out with NOEMA and the SMA. The HNC(1–0)/HCN(1–0) line ratio in SDP.9 suggests the presence of photon-dominated regions, as happens in most local (U)LIRGs. The CO, HCN, and HCO+ spectral line energy distribution (SLEDs) of SDP.9 are compatible to those found for many local, IR-bright galaxies, indicating that the molecular gas in local and high-redshift dusty starbursts can have similar excitation conditions. We obtain that the correlation between total IR ({L}{IR}) and dense line ({L}{dense}) luminosity in SDP.9 and SDP.11 and local star-forming galaxies can be represented by a single relation. We argue that the scatter of the {L}{IR}{--}{L}{dense} correlation, together with the lack of sensitive dense molecular gas tracer observations for a homogeneous sample of high-redshift galaxies, prevents us from distinguishing differential trends with redshift. Our results suggest that the intense star formation found in some high-redshift, dusty, luminous starbursts is associated with more massive dense molecular gas reservoirs and higher dense molecular gas fractions.